Intelligence is ability to predict and plan (selfpredict), which
can only be done by discovering and projecting patterns. This perspective is well
established: pattern recognition is a core of any IQ test. But there is no general
and constructive definition of either pattern or recognition (quantified similarity).
So, I came up with my own definitions, which directly translate into algorithm proposed
below.
For excellent popular introductions to cognitionasprediction
thesis see “On Intelligence” by Jeff Hawkins and “How to Create a Mind“ by Ray Kurzweil. But on a technical level, they
and most current researchers implement pattern discovery via artificial neural
networks, which operate in a very coarse statistical fashion.
Less coarse (more selective) are Capsule Networks, recently introduced by
Geoffrey Hinton et al. But they are largely ad hock, still workinprogress,
and depend on layers of CNN. Neither ANN nor CapsNet
is theoretically derived. I outline my approach below and then compare it to
ANN, biological NN, CapsNet, and clustering.
We need help with design and implementation of
this algorithm: open project CogAlg on GitHub,
in Python. There is a prize for contributions, or monthly payment if you have
good track record, see the last part here. Contributions should be justified in
terms of strictly incremental search for similarity, forming hierarchical patterns.
These terms are defined below, but if you have better definitions, that would
be even more valuable.
This content is published under Creative Commons
Attribution 4.0 International License.
Outline of my approach
Proposed algorithm is a clean design for deep
learning: nonneuromorphic,
substatistical, comparisonfirst.
It performs hierarchical search
for patterns, by crosscomparing inputs over selectively incremental distance
and composition. “Incremental” means that firstlevel inputs must be minimal in
complexity, such as pixels of video or equivalents in other modalities. Symbolic
data is secondhand, it shouldn’t be used as primary input.
Their comparison must also
be minimal in complexity: a lossless transform by inverse arithmetic operations.
Initial comparison is by subtraction, similar to edge detection kernel in CNN. But my comparison forms partial match along with miss,
and accumulates both inside patterns: spans of samesign miss or match deviation.
Match is compression of represented magnitude by replacing larger comparand
with the miss.
Specific match and miss between
two variables are determined by the power of comparison operation:
 comparison by subtraction
increases match to a smaller comparand and reduces miss to a difference,
 comparison by division
increases match to a multiple and reduces miss to a fraction, and so on, see part
1.
These comparisons form patterns:
representations of input spans with constant sign of inputtofeedback miss.
Search hierarchy has two
orders of feedback: within and between levels, forming lateral and vertical
patterns. Lateral feedback is prior inputs, and their comparison forms
difference patterns: spans of inputs with increasing or decreasing magnitude. Vertical
feedback is average higherlevel match, and comparison forms predictive value
patterns: spans of inputs with above or below average match. This feedback is
restricted to match: higher order of representation, to justify redundancy of value
patterns to lateral difference patterns.
Higherlevel inputs are
patterns formed by lowerlevel comparisons. They represent results or
derivatives: match and miss per compared input parameter. So, number of parameters
per pattern is selectively multiplied on each level. Match and miss between
patterns are combined matches or misses between their parameters. To maximize selectivity, search must be strictly incremental in
distance, derivation, and composition over both. Which implies a unique set of
operations per level of search, hence a singular in “cognitive algorithm“.
Resulting hierarchy is a dynamic
pipeline: terminated patterns are outputted for comparison on the next level, hence
a new level must be formed for pattern terminated by current top level. Which continues
as long as system receives novel inputs. As distinct from autoencoders (current
mainstay in unsupervised learning), there is no need for decoding: comparison
is done on each level, whose output is also fed back to filter lower levels. My
comparison is a form of inference, and feedback of summed miss to update
filters is a form of training.
To discover anything
complex at “polynomial” cost, resulting patterns should also be hierarchical.
In my model, each level of search adds one level of composition and one
sublevel of differentiation to each input pattern.
Higherlevel search is
selective per level of resulting pattern. Both composition and selection
speedup search, to form longer range spatiotemporal and then conceptual
patterns. Which also send feedback: filters and then motor action, to select
lowerlevel inputs and locations with aboveaverage additive predictive value
(part 3).
Hierarchical approaches are
common in unsupervised learning, and all do some sort of
pattern recognition.
But none that I know of is
strictly incremental in scope and complexity of discoverable patterns. Which is necessary for selectivity, thus scalability, vs.
combinatorial explosion in search space. But selection is very
expensive upfront and won’t pay in simple test problems. So, it’s not suitable for immediate experimentation. This is
probably why no one else seems to be working on anything sufficiently similar
to my algorithm.
Autonomous cognition must
start with analog inputs, such as video or audio. All symbolic data (any kind
of language) is encoded by some prior cognitive process. To discover meaningful patterns in symbols, they must be decoded
before being crosscompared. And the difficulty of decoding is exponential
with the level of encoding, thus hierarchical learning starting with raw
sensory input is by far the easiest to implement (part 0).
Such raw inputs have modalityspecific
properties and comparison should be adjusted accordingly, by feedback or manually.
For example, vision relies on reflected light: brightness or albedo don’t
directly represent source of impact, though they do represent some resistance
to a very "light" impact. Uniformity of albedo indicates some common
property within object, so it should form patterns. But the degree of
commonality doesn’t depend on intensity of albedo, so match should be defined
indirectly, as belowaverage difference or ratio of albedo.
Many readers see a gap between my theory and algorithm, or a lack of
the latter. It’s true that algorithm is far from complete, but we are well on our
way in translating the aboveexplained principles into code: https://github.com/boriskz/CogAlg.
Final algorithm will be a metalevel of search: 1st level + recursive
complexity increment to generate incrementally higher levels. Our universe is a
spacetime continuum, thus a level is a 3D or 4D cycle. Others note lack of
higher math, but that’s because it's not selectively incremental.
Comparison to artificial
and biological neural networks
ANN learns via some version of Hebbian “fire together, wire together” coincidence reinforcement.
Normally, “neuron’s” inputs are weighed at synapses, then summed and
thresholded into output. Final output also back propagates to synapses
and is combined with their last input to adjust the weights. This weight
adjustment is learning. But prior summation degrades resolution of the inputs, precluding any comparison between them (original inputs are
recoverable, but why degrade and then spend a ton of computation restoring
them).
This is a statistical
method: inputs are summed within samples defined by initially random weights. Weighted
inputs propagate through hidden layers of ANN, then are compared to toplayer
template, forming an error. That error then backpropagates to train weights
into meaningful values by Stochastic Gradient Descent.
So, this process is
comparisonlast, vs. my comparisonfirst. Which makes it too coarse to scale
without supervision or taskspecific reinforcement, and it gets exponentially more
coarse as ANNs become deeper. Due to lossiness of intermediate summation, this forward
/ back cycle must be repeated tens of thousands of times.
Another problem is that SGD
minimizes error (miss), which doesn’t always correlate with maximizing match. Cognition
builds a predictive model of environment, value of representations in this
model must be positive. But error is primarily negative, so the value of reducing
it can’t be directly combined with the value of extending training. CogAlg
maximizes compression, which increases representational and predictive power of
the system. Nothing is randomized and resulting patterns are immediately meaningful,
although initially tentative.
Inspiration by the brain
kept ANN research going for decades before they became useful. But their
“neurons” are mere stick figures for real ones. Of course, most of complexity
in a neuron is due to constraints of biology. Ironically, weighted summation in ANN may also be a nolonger needed
compensation for such constraint:
neural memory requires
dedicated connections (synapses), which makes individual
input comparison very expensive. But not anymore, we now have dirtcheap randomaccess
memory.
Other biological
constraints are very slow neurons, and the imperative of fast reaction for
survival in the wild. Both favor fast though crude summation (vs.
slower onetoone comparison), at the cost of glacial training. Reaction speed
became less important: modern society is quite secure, while continuous
learning is far more important because of accelerating progress.
Another likely reason for the
prevalence of summation in neurons is to reduce noise, because they often fire
at random. That’s probably how they initially connect in the womb, and then
temporarily maintain latent connections. But none of that is relevant for electronic
circuits.
In general, distribution of
computation across physical network is only justifiable for nodes that are complex
enough for the cost of connections. Simpler constructs should be in local
memory, which computers have but brain doesn’t. Cognitive function is a search
for patterns, so parameters that characterize patterns should be stored locally.
Increasing speed by parallelized crosscomparison among patterns is a secondary
matter.
Comparison to Capsule
Networks
The nearest experimentally
successful method is recently introduced “capsules”. Some similarities to CogAlg:
 capsules also output
multivariate vectors, “encapsulating” several properties, similar to my
patterns,
 these properties also
include coordinates and dimensions, compared to compute differences and ratios,
 these distances and
proportions are also compared to find “equivariance” or affine transformations,
 capsules also send direct
feedback to lower layer (dynamic routing), vs. transhiddenlayer backprop in
ANN
But measure of similarity
in CapsNet (“agreement” in dynamic routing) is an unprincipled dot product, vs.
additive compression in CogAlg. This is very common, most recognition algorithms
use matrix multiplication. Conceptually, similarity is a common subset of
comparands, and multiplication vastly exaggerates it by forming a superset. Exaggeration
adds resistance to noise, but at the cost of drastically impaired precision. Distinction
between input and noise is casespecific, it should be learned from the input
itself, not built in the algorithm.
Common subset of two
integers is the smaller of them, = compression of represented magnitude by replacing
larger input with the difference between inputs. This is a direct implication
of information theory: compression must be a measure of similarity, but no one
else seems to use it from the bottom up. It’s not sufficient per se, basic working
measure would probably be more complex, but minimum is unavoidable as a
starting point.
Some other problems I have
with current implementation of CapsNet:
 CapsNet is initially
fullyconnected, with a networkcentric bias toward uniform matrix operations (vs.
conditional unfolding). In CogAlg, search is selective over incremental
dimensionality and distance.
 they use CNN for initial layers
to recognize basic features, but a truly general method should apply the same
principles on all levels of processing, any differentiation should be learned
rather than builtin.
 capsules of all layers contain
the same parameters: probability and pose variables, while I think the number of parameters should
be incremental with elevation, each level forms higherorder derivatives of
input variables.
 the number of layers is fixed,
while I think it should be incremental with experience.
My patterns have match
instead of probability, a miss that includes pose variables, plus selected properties
of lowerlevel patterns. In my terms, Hinton’s equivariance is a match between misses:
differences and distances.
All these variables are derived
by incrementally complex comparison: core operation on all levels of CogAlg.
My hierarchy is also
dynamic: pattern is displaced from level by a miss to new input, then forwarded
to existing or newly formed higher level. So, higherlevel patterns include
lowerlevel variables, as well as their derivatives. The derivatives are summed
within pattern, then evaluated for extending intrapattern search and feedback.
Thus, both hierarchy of patterns per system and subhierarchy of variables per
pattern expand with experience.
Comparison to conventional clustering
techniques
My approach is a form of hierarchical clustering, but match in conventional clustering is
inverted distance: a misleading term for difference. This is the opposite of multiplication
between comparands, which computes similarity (match) but no coincident
difference (miss). I believe both should be computed because each has
independent predictive value: match is a common subset, distinct from and
complementary to the difference.
This distinction is not
apparent in modalities where signal carrier is “light” and reflected, such as
visual input. There, magnitude (brightness) of input parameter or its match
(compression of input magnitude) has low correlation with predictive value. This
is true for most raw information sources, but match is a key higherorder
parameter. That is, match of parameters that do represent predictive value (such
as inverted distance), should be a criterion / metrics for higherlevel
clustering of patterns that contain / encapsulate them.
Again, main feature of my
approach is incrementally deep hierarchical syntax (encapsulated parameters) of
my patterns. Which means that metrics will change with elevation: criterion of
higherlevel clustering will be derived from comparison of lowerlevel parameters
between their patterns. I can’t find an analogue to this evolving hierarchical nature
of both elements and metrics in any other clustering technique.
elaboration below, some out
of date:
0. Cognition vs. evolution, analog vs. symbolic initial input
1. Comparison: quantifying match and miss between two variables
2. Forward search and patterns, implementation for image recognition
in video
3. Feedback of filters, attentional input selection, imagination,
motor action
4. Initial levels of search, corresponding orders of feedback, and
resulting patterns:
level 1: comparison to past
inputs, forming difference and relative match patterns
level 2: additional
evaluation of resulting patterns for feedback, forming filter patterns
level 3: additional evaluation
of projected filter patterns, forming updatedinput patterns
5. Comparison between variable types within a pattern
6. Cartesian dimensions and sensory modalities
7. Notes on clustering, ANNs, and probabilistic inference
8. Notes on working mindset, “hiring” and a prize for contributions
0. Cognition vs. evolution,
analog vs. symbolic initial input
Some say intelligence can
be recognized but not defined. I think that’s absurd: we recognize some
implicit definition. Others define intelligence as a problemsolving ability,
but the only general problem is efficient search for solutions. Efficiency is a
function of selection among inputs, vs. bruteforce alltoall search. This
selection is by predicted value of the inputs, and prediction is interactive
projection of their patterns. Some agree that intelligence is all about pattern
discovery, but define pattern as a crude statistical coincidence.
Of course, the only
mechanism known to produce humanlevel intelligence is even cruder, and that
shows in haphazard construction of our brains. Algorithmically simple, biological
evolution alters heritable traits at random and selects those with
aboveaverage reproductive fitness. But this process requires almost
inconceivable computing power because selection is extremely coarse: on the
level of whole genome rather than individual traits, and also because
intelligence is only one of many factors in reproductive fitness.
Random variation in
evolutionary algorithms, generative RBMs, and so on, is antithetical to intelligence. Intelligent
variation must be driven by feedback within cognitive hierarchy: higher levels
are presumably “smarter” than lower ones. That is, higherlevel inputs
represent operations that formed them, and are evaluated to alter future
lowerlevel operations. Basic operations are comparison and summation among
inputs, defined by their range and resolution, analogous to reproduction in
genetic algorithms.
Range of comparison per
conservedresolution input should increase if projected match (cognitive
fitness function) exceeds average match per comparison. In any nonrandom
environment, average match declines with the distance between comparands. Thus,
search over increasing distance requires selection of above average
comparands. Any delay, coarseness, and inaccuracy of such selection is
multiplied at each search expansion, soon resulting in combinatorial explosion
of unproductive (low additive match) comparisons.
Hence, my model is strictly
incremental: search starts with minimalcomplexity inputs and expands with
minimal increments in their range and complexity (syntax). At each level, there
is only one best increment, projected to discover the greatest additive match.
No other AGI approach follows this principle.
I guess people who aim for
humanlevel intelligence are impatient with small increments and simple sensory
data. Yet, this is the most theoretical problem ever, demanding the longest
delay in gratification.
symbolic obsession and its
discontents
Current Machine Learning and
related theories (AIT, Bayesian inference, etc.) are largely statistical also because
they were developed primarily for symbolic data. Such data, precompressed and
preselected by humans, is far more valuable than sensory inputs it was
ultimately derived from. But due to this selection and compression, proximate
symbols are not likely to match, and partial match between them is very hard to
quantify. Hence, symbolic data is a misleading initial target for developing
conceptually consistent algorithm.
Use of symbolic data as
initial inputs in AGI projects betrays profound misunderstanding of cognition.
Even children, predisposed to learn language, only become fluent after years of
directly observing things their parents talk about. Words are mere labels for
concepts, the most important of which are spatiotemporal patterns, generalized
from multimodal sensory experience. Topdown reconstruction of such patterns
solely from correlations among their labels should be exponentially more
difficult than their bottomup construction.
All our knowledge is
ultimately derived from senses, but lower levels of human perception are
unconscious. Only generalized concepts make it into our consciousness, AKA declarative memory, where we assign them symbols (words) to
facilitate communication. This brainspecific constraint creates heavy symbolic
vs. subsymbolic bias, especially strong in artificial intelligentsia. Which is
putting a cart in front of a horse: most words are meaningless unless coupled
with implicit representations of sensory patterns.
To be incrementally
selective, cognitive algorithm must exploit proximity first, which is only
productive for continuous and losstolerant raw sensory data. Symbolic
data is already compressed: consecutive characters and words in text won’t
match. It’s also encoded with distant crossreferences, that are hardly ever
explicit outside of a brain. Text looks quite random unless you know the code:
operations that generalized pixels into patterns (objects, processes,
concepts). That means any algorithm designed specifically for text will not be
consistently incremental in the range of search, which will impair its
scalability.
In Machine Learning, input
is string, frame, or video sequence of a defined length, with artificial
separation between training and inference. In my approach, learning is
continuous and interactive. Initial inputs are streamed pixels of maximal
resolution, and higherlevel inputs are multivariate patterns formed by
comparing lowerlevel inputs. Spatiotemporal range of inputs, and selective
search across them, is extended indefinitely. This expansion is directed by
higherlevel feedback, just as it is in human learning.
Everything ever written is
related to my subject, but nothing is close enough: not other method is meant
to be fully consistent. Hence a dire scarcity of references here. My approach
is presented bottomup (parts 1  6), thus can be understood without
references. But that requires a clean context,  hopefully cleaned out by
reader‘s own introspective generalization. Other (insufficiently) related
approaches are addressed above and in part 7. I also have a more advanced workinprogress,
but will need a meaningful feedback to elaborate.
1. Comparison: quantifying
match and miss between two variables
First of all, we must
quantify predictive value. Algorithmic information theory defines it as compressibility of representation.
Which is perfectly fine, but compression is currently computed only for sequences
of inputs.
To enable far more
incremental selection, thus scalable search, I quantify match between
individual inputs. Partial match is a finer dimension of analysis, vs. binary
same  different instances. This is similar to the way probabilistic inference
improved on classical logic, by quantifying probability vs. binary true  false
values.
I define match as a
complementary of miss. Basic miss is a difference between comparands, hence
match is a smaller comparand. In other words, match is a compression of larger
comparand’s magnitude by replacing it with the difference to smaller comparand.
Ultimate criterion is recorded magnitude, rather than bits of memory it
occupies, because the former represents physical impact that we want to
predict. The volume of memory used to record that magnitude depends on prior
compression, which is not an objective parameter.
This definition is
tautological: smaller input is a common subset of both inputs, = sum of AND between their uncompressed (unary code) representations. Some may
object that match includes the case when both inputs equal zero, but then match
also equals zero. The purpose is prediction, which is a representational
equivalent of conservation in physics. Ultimately, we’re predicting some
potential impact on observer, represented by input. Zero input means zero
impact, which has no conservable inertia, thus no intrinsic predictive value.
With incremental
complexity, initial inputs have binary resolution and implicit shared
coordinate (as a macroparameter, resolution of coordinate lags that of an input).
Compression of bit inputs by AND is well known as digitization: substitution of
two lower 1 bits with one higher 1 bit. Resolution of coordinate (input
summation span) is adjusted by feedback to form integers that are large enough to
produce aboveaverage match.
Nextorder compression is
comparison between consecutive integers, with binary (before  after)
coordinate. Basic comparison is inverse arithmetic operation of incremental
power: AND, subtraction, division, logarithm, and so on. Additive match is
achieved by comparison of a higher power than that which produced comparands:
comparison by AND will not further compress integers previously digitized by
AND.
Rather, initial comparison
between integers is by subtraction, resulting difference is miss, and smaller
input is absolute match. For example, if inputs are 4 and 7, then miss
is 3, and their match or common subset is 4. Difference is smaller than XOR (nonzero
complementary of AND) because XOR may include oppositesign (oppositedirection)
bit pairs 0, 1 and 1, 0, which are cancelledout by subtraction.
Comparison by division
forms ratio, which is a compressed difference. This
compression is explicit in long division: match is accumulated over iterative
subtraction of smaller comparand from remaining difference. In other words,
this is also a comparison by subtraction, but between different orders of
derivation. Resulting match is smaller comparand * integer part of ratio, and miss is final reminder or
fractional part of ratio.
Ratio can be further
compressed by converting it to radix or logarithm, and so on.
By reducing miss,
higherpower comparison increases complementary match (match = larger input 
miss):
to be compressed: larger
input  XOR  difference: combined currentorder match &
miss
additive match:
AND

oppositesign XOR  multiple: of a smaller input within a
difference
remaining miss: XOR

difference 
fraction: complementary to multiple
within a ratio
But the costs of operations
and record of incidental sign, fraction, irrational fraction, etc. may grow
even faster. To justify these costs, power of comparison should only increase
for inputs sufficiently compressed by prior order of comparison: AND for bit
inputs, SUB for integer inputs, DIV for pattern inputs, etc.
Basic selection criterion is relative
match: current match  past match cooccurring with average higherlevel
projected match. Such past match is a filter that determines inclusion of the
input into positive or negative (above or below average) predictive value
pattern. Relative match is accumulated
until it exceeds the cost of updating lowerlevel filter, which terminates filter
pattern (samefilter input span) and initializes a new one.
Compression also depends on
resolution of coordinate (default input summation span), and on resolution of
input magnitude. Projected match must be kept above system’s average by adjusting
corresponding resolution filters: most significant bits and least significant
bits of both coordinate and magnitude.
Separate filters are formed for
each type of compared variable. For example, magnitude of brightness or even of
albedo is not very predictive, because perceived light is very lowimpact. Thus,
primary filter should increase, reducing total span and cost of positive value
patterns, possibly down to 0 (more on that below).
2. Forward search and
patterns, implementation for image recognition in video
Pattern is a contiguous
span of inputs that form aboveaverage matches, similar to conventional cluster.
As explained above, matches
and misses (derivatives) are produced by comparing consecutive inputs. These
derivatives are summed within a pattern and then compared between patterns on
the next level of search, adding new derivatives to a higher pattern. Patterns
are defined contiguously on each level, but positive and negative patterns are
always interlaced, thus nextlevel samesign comparison is discontinuous.
Negative patterns represent
contrast or discontinuity between positive patterns, which is a one or higher
dimensional equivalent of difference between zerodimensional pixels. As with
differences, projection of a negative pattern competes with projection of
adjacent positive pattern. But match and difference are derived from the
same input pair, while positive and negative patterns represent separate
spans of inputs.
Negative match patterns are
not predictive on its own but are valuable for allocation: computational
resources of nolonger predictive pattern should be used elsewhere. Hence,
the value of negative pattern is borrowed from predictive value of coprojected
positive pattern, as long as combined additive match remains above average. Consecutive
positive and negative patterns project over same future input span, and these
projections partly cancel each other. So, they should be combined to form
feedback, as explained in part 3.
Initial match is evaluated
for inclusion into higher positive or negative pattern. The value is summed
until its sign changes, and if positive, evaluated again for crosscomparison
among constituent inputs over increased distance. Second evaluation is
necessary because the cost of incremental syntax generated by crosscomparing
is per pattern rather than per input. Pattern is terminated and
outputted to the next level when value sign changes. On the next level, it is
compared to previous patterns of the same compositional order.
Initial inputs are pixels
of video, or equivalent limit of positional resolution in other modalities. Hierarchical
search on higher levels should discover patterns representing empirical objects
and processes, and then relational logical and mathematical shortcuts,
eventually exceeding generality of our semantic concepts.
In cognitive terms, everything
we know is a pattern, the rest of input (noise) is filtered out by perception.
For online learning, all levels should receive inputs from lower levels and
feedback from higher levels in parallel.
spacetime dimensionality
and initial implementation
Any prediction has two
components: what and where. We must have both: value of prediction = precision
of what * precision of where. That “where” is currently neglected: statistical
ML represents spacetime at greatly reduced resolution, if at all. In the brain
and some neuromorphic models, “where” is represented in a separate network.
That makes transfer of positional information very expensive and coarse, reducing
predictive value of representations. There is no such separation in my patterns,
they represent both what and where as local vars.
My core algorithm is 1D:
time only (part 4). Our spacetime is 4D, but each of these dimensions
can be mapped on one level of search. This way, levels can select input
patterns that are strong enough to justify the cost of representing additional
dimension, as well as derivatives (matches and differences) in that dimension.
Initial 4D cycle of search
would compare contiguous inputs, similarly to connectedcomponent analysis:
level 1 compares
consecutive 0D pixels within horizontal scan line, forming 1D patterns: line
segments.
level 2 compares contiguous
1D patterns between consecutive lines in a frame, forming 2D patterns: blobs.
level 3 compares contiguous
2D patterns between incrementaldepth frames, forming 3D patterns: objects.
level 4 compares contiguous
3D patterns in temporal sequence, forming 4D patterns: processes.
(in
simple video, time is added on level 3 and depth is computed from derivatives)
Subsequent cycles would
compare 4D input patterns over increasing distance in each dimension, forming
longerrange discontinuous patterns. These cycles can be coded
as implementation shortcut, or form by feedback of core algorithm itself, which
should be able to discover maximal dimensionality of inputs.
“Dimension” here is parameter that defines external sequence and distance among
inputs. This is different from conventional clustering,
were both external and internal parameters are dimensions. More in part 6.
However,
average match at a given distance in our spacetime is presumably equal over
all four dimensions. That means patterns defined in fewer dimensions will be fundamentally
limited and biased by the angle of scanning. Hence, initial pixel comparison and
inclusion into patterns should also be over 4D at once, or at least over 2D for
images and 3D for video. This would be a universespecific extension of my core
algorithm.
There is also a visionspecific
adaptation in the way I define initial match:
Predictive visual property is
albedo, which means locally stable ratio of brightness. Since lighting is
usually uniform over much larger area than pixel, the difference between
adjacent pixels should also be stable.
Similar brightness indicates
match of some underlying property, so it should form match patterns. But all values of brightness are roughly equally
predictive, thus match can’t be defined as min brightness.
Comparison also forms difference,
and low difference implies stability
(compressibility) of input. So, match between pixels is defined as average_difference  difference. On higher levels, this inverse difference can be used as a
selection criterion, with match computed directly as a minimum or common
subset. Initial
difference across space is a product of past interactions in time (time is a macrodimension
due to a lower rate of change). Thus, difference or match across space is
predictive of change or stability over time.
We are currently working on code
for 1^{st} level algorithm: https://github.com/boriskz/CogAlg/wiki
1D code is complete, we are extending
it to 2D for image clustering and recognition, and then to 3D (video) for object
and process recognition. Higher levels for each will process discontinuous search
among fullD patterns. We will then add colors, maybe audio and text. Initial
testing could be recognition of labeled images, but 2D is a poor representation
of our 4D world, video or stereo video should be far better.
Complete hierarchical algorithm
will have twolevel code:
 1^{st} level algorithm:
contiguous crosscomparison over fullD cycle, plus bitfilter feedback
 recursive increment in
complexity of both crosscomparison and feedback, generating nextlevel
algorithm.
3. Feedback of filters,
attentional input selection, imagination, motor action
(needs work)
After evaluation for
inclusion into higherlevel pattern, input is also evaluated as feedback to
lower levels. Feedback is update to filters that evaluate forward (Î›) and
feedback (V), as described above but on
lower level.
Basic filter is average
value of input’s projected match that cooccurs with (thus predicts) average
higherlevel match within a positive (aboveaverage) pattern. Both values are
represented in resulting patterns.
Feedback value = forward
value  value of derivatives /2, both of an input pattern. In turn, forward
value is determined by higherlevel feedback, and so on. Thus, all higher
levels affect selection on lowerlevel inputs. This is because the span of each
pattern approximates, hence projects over, combined span of all lower levels.
Indirect feedback
propagates levelsequentially, more expensive shortcut feedback is sent to
selected levels.
Negative derivatives
project increasing match: match to subsequent inputs is greater than to previous inputs.
Such feedback will reduce
lowerlevel filter. If filter is zero, all inputs are crosscompared, and if
filter is negative, it is applied to cancel subsequent filters for incrementally
longerrange crosscomparison.
There is one filter for
each compared variable within input pattern, initialized at 0 and
updated by feedback.
novelty vs. generality
Any integrated system must
have a common ultimate selection criterion. Two obvious cognitive criteria are
novelty and generality: miss and match. But we can’t select for both, they
exhaust all possibilities. Novelty can’t be primary criterion: it would select
for noise and filter out all patterns, which are defined by match. On the other
hand, to maximize match of inputs to memory we can stare at a wall: lock into
predictable environments. But of course, natural curiosity actively skips
predictable locations, thus reducing the match.
This dilemma is resolved if
we maximize predictive power: projected match, rather than actual match, of inputs to records. To the extent that new
match was projected by past inputs, it doesn’t add to their projected match.
But neither does noise: novelty (difference to past inputs) that is not
projected to persist (match) in the future.
Additive projected match =
downward (V) match to subsequent inputs  upward (Î›) match to previous inputs.
This Î›V asymmetry comes from projecting difference or
value (both are signed) of input relative to feedback.
Î› match is initially
projected from match between input and its higherlevel average. Such averages
may be included in feedback, along with average match (filter). Each
average is compared to sametype variable of an input, and resulting match (redundancy to a higher level) is subtracted from input
match before evaluation.
Averages represent past vs.
current higher level and should be projected over feedback delay:
average += average
difference * (delay / average span) /2. Value of inputtoaverage
match is reduced by higherlevel match rate: rM= match / input, so
additive match = input match  inputtoaverage match * rM.
Adjustment by
inputtoaverage match is done if rM is aboveaverage for incurred cost of
processing.
So, basic selection for
novelty is subtraction of adjusted Î› matchtofilter. V selection for generality, on the other hand, is a function of
backprojected difference of match, which is a higherderivation feedback.
Projected increase of match adds to, and decrease of match subtracts from,
predictive value of future inputs.
Higherorder selection for
novelty should skip (or avoid processing) future input spans predicted with
high certainty. This is a selection of projected vs. actual inputs, covered in
part 4, level 3.
More generally, Î›V projection asymmetry is expressed by
differences of incremental representation order:
difference in magnitude of initial inputs: projected next input = last input +
difference/2,
difference in input match, a subset of magnitude: projected next match = last match + match
difference/2,
difference in match of match, a subsubset of magnitude, projected correspondingly, and so on.
Ultimate criterion is top
order of match on a top level of search: the most predictive parameter in a
system.
imagination, planning,
action
Imagination is not actually
original, it can only be formalized as interactive projection of known
patterns. Strong patterns send positive feedback to lowerlevel locations where
they are projected to reoccur at higher resolution, to increase represented
detail. When originally distant patterns are projected to reoccur in the same
location, their projections interact and combine. This is a generative (vs.
reductive) mechanism.
Interaction is comparison
and summation between sametype variables of coprojected patterns: direct
feedback from multiple higherlevel patterns levels to the same external to
them location. Location is a span of search within a lower level, which
receives feedback with matching projected coordinates. Evaluation of actual
inputs is delayed until additive match projected in their location exceeds the
value of combined filters.
Comparison forms patterns
of filters to compress their representations and selectively project their
match, same as with original inputs. Summation of their differences cancels or
reinforces corresponding variables, forming combined filter, including
secondary patterns of repulsion or attraction. These patterns per variable are
discovered on a higher level, by comparing past inputs with their
multiplefeedback projections.
Combined filter is preevaluated:
projected value of positive patterns is compared to projected cost of
evaluating all inputs, both within a filtered location. Resulting prevalue
(value of evaluation) is negative when the latter exceeds the former: projected
inputs are not worth evaluating and their span / location is skipped.
Iterative skipping of input
spans is the most basic motor feedback: it increments coordinates and
differences of the filters. Crossfilter search then proceeds over multiple
“imagined” locations, before finding one with projected aboveaverage additive
match. That’s where skipping stops and actual inputs are received.
Cognitive component of
action is planning. It’s a kind of imagination where projected patterns include
those that represent system itself. Feedback of such selfpatterns eventually
reaches the bottom of representational hierarchy: sensors and actuators,
adjusting their sensitivity and coordinates. Adjustment of spatial coordinates (filters)
is action. Such environmental interface is part of any cognitive system,
although actuators are optional.
4. Initial levels of
search, corresponding orders of feedback and resulting patterns
This part recapitulates and
expands on my core algorithm, which operates in one dimension: time only.
Spatial and derived dimensions are covered in part 6. Even within 1D, search is
hierarchical in scope, containing any number of levels. New level is added when current top level terminates and outputs
the pattern it formed.
Higherlevel patterns are
fed back to select future inputs on lower levels. Feedback is sent to all lower
levels because span of each pattern approximates combined span of inputs within
whole hierarchy below it.
So, deeper hierarchy forms
higher orders of feedback, with increasing elevation and scope relative to its
target: samelevel prior input, higherlevel match average,
beyondthenextlevel match value average, etc.
These orders of feedback
represent corresponding order of input compression: input, match
between inputs, match between matches, etc. Such compression is produced by
comparing inputs to feedback of all orders. Comparisons form patterns, the type
of which corresponds to relative span of compared feedback:
1: prior inputs are compared to the following ones on the same level,
forming difference patterns dPs,
2: higherlevel match is used to evaluate match between inputs, forming value
patterns vPs,
3: higherhierarchy value revaluates positive values of match, forming more selective
shortcut patterns sPs
Feedback of 2^{nd} order consists of input filters (if) defining value patterns, and coordinate filters
(Cf) defining positional
resolution and relative distance to future inputs.
Feedback of 3^{rd} order is shortcut filters for beyondthenext level. These
filters, sent to a location defined by attached coordinate filters, form
higherorder value patterns for deeper internal and distantlevel comparison.
Higherorder patterns are
more selective: difference is as likely to be positive as negative, while value
is far more likely to be negative, because positive patterns add costs of
reevaluation for extended crosscomparison among their inputs. And so on, with
selection and reevaluation for each higher order of positive patterns.
Negative patterns are still compared as a whole: their weak match is
compensated by greater span.
All orders of patterns
formed on the same level are redundant representations of the same inputs.
Patterns contain representation of match between their inputs, which are
compared by higherorder operations. Such operations increase overall match by
combining results of lowerorder comparisons across pattern’s variables:
0Le: AND of bit
inputs to form digitized integers, containing multiple powers of two
1Le: SUB of
integers to form patterns, over additional external dimensions = pattern length
L
2Le: DIV of
multiples (L) to form ratio patterns,
over additional distances = negative pattern length LL
3Le: LOG of
powers (LLs), etc.
Starting from second level, comparison is selective per element of an
input.
Such power increase also
applies in comparison to higherorder feedback, with a lag of one level per
order.
Power of coordinate filters
also lags the power of input filters by one level:
1Le fb: binary
sensor resolution: minimal and maximal detectable input value and coordinate
increments
2Le fb:
integervalued average match and relative initial coordinate (skipping
intermediate coordinates)
3Le fb:
rationalvalued coefficient per variable and multiple skipped coordinate range
4Le fb:
realvalued coefficients and multiple coordinaterange skip
I am defining initial
levels to find recurring increments in operations per level, which could then
be applied to generate higher levels recursively, by incrementing syntax of
output patterns and of feedback filters per level.
operations per generic
level (out of date)
Level 0 digitizes inputs, filtered by minimal
detectable magnitude: least significant bit (i LSB). These bits are AND compared, then their
matches are AND compared again, and so on, forming integer
outputs. This is identical to iterative summation and bitfiltering by
sequentially doubled i LSB.
Level 1 compares consecutive integers, forming ± difference patterns (dP s). dP s are then evaluated to crosscompare their
individual differences, and so on, selectively increasing derivation of
patterns.
Evaluation: dP M (summed match)  dP aM (dP M per average match between
differences in level 2 inputs).
Integers are limited by the
number of digits (#b), and input span: least
significant bit of coordinate (C LSB).
No 1^{st} level feedback: fL cost is additive to dP cost, thus must be justified by the value of dP (and coincident difference in value of patterns
filtered by adjusted i LSB), which is not known till dP is outputted to 2^{nd} level.
Level 2 evaluates match within dP s  bf L (dP) s, forming ± value patterns: vP s  vP (bf L) s. +vP s are evaluated for crosscomparison of their dP s, then of resulting
derivatives, then of inputted derivation levels. +vP (bf L) s are evaluated to crosscompare bf L s, then dP s, adjusted by the difference between their bit filters, and so
on.
dP variables are compared by
subtraction, then resulting matches are combined with dP M (match
within dP) to evaluate
these variables for crosscomparison by division, to normalize
for the difference in their span.
// match filter is also normalized by span ratio
before evaluation, samepower evaluation and comparison?
Feedback: input dP s  bf L (dP) are backprojected and resulting magnitude is evaluated to
increment or decrement 0^{th} level i LSB. Such increments terminate bitfilter span ( bf L (dP)), output it to 2^{nd} level,
and initiate a new i LSB span to filter future
inputs. // bf L (dP) representation: bf , #dP, Î£ dP, Q (dP).
Level 3 evaluates match in input vP s or f L (vP) s, forming ± evaluationvalue patterns: eP s  eP (fL) s. Positive eP s are evaluated for crosscomparison of their vP s ( dP s ( derivatives ( derivation levels ( lower searchlevel sources: buffered or external locations
(selected sources may directly specify strong 3^{rd} level subpatterns).
Feedback: input vP is backprojected, resulting match is compared to 2^{nd} level filter, and the difference is evaluated vs. filterupdate
filter. If update value is positive, the difference is
added to 2^{nd} level filter, and filter span is terminated.
Same for adjustment of previously covered bit filters and 2^{nd} level filterupdate filters?
This is similar to 2^{nd} level operations, but input vP s are separated by skippedinput spans. These
spans are a filter of coordinate (Cf, higherorder than f for 2^{nd} level inputs), produced by prevaluation of future inputs:
projected novel match =
projected magnitude * average match per magnitude  projectedinput match?
Prevalue is then evaluated
vs. 3^{rd} level evaluation filter + lowerlevel
processing cost, and negative prevaluevalue input span (= span of backprojecting
input) is skipped: its inputs are not processed on lower levels.
// no prevaluation on 2^{nd} level: the cost is higher than potential savings of only 1^{st} level processing costs?
As distinct from input
filters, Cf is defined individually rather than per filter
span. This is because the cost of Cf update: span representation
and interruption of processing on all lower levels, is minor compared to the
value of represented contents? ±eP = ±Cf: individual skip evaluation, no flushing?
or interruption is
predetermined, as with Cb, fixed C f within C f L: a span of sampling across fixedL gaps?
alternating signed Cf s are averaged ±vP s?
Division: between L s, also inputs within
minimaldepth continuous dsign or morder derivation hierarchy?
tentative generalizations
and extrapolations
So, filter resolution is
increased per level, first for i filters and then for C filters: level 0 has input bit filter,
level 1 adds coordinate bit filter, level 2 adds input integer filter, level 3
adds coordinate integer filter.
// coordinate filters (Cb, Cf) are not inputspecific,
patterns are formed by comparing their contents.
Level 4 adds input multiple filter: eP match and its derivatives, applied in parallel to corresponding
variables of input pattern. Variablevalues are multiplied and evaluated to
form patternvalue, for inclusion into nextlevel ±pattern // if separately evaluated, inputvariable value = deviation from
average: signreversed match?
Level 5 adds coordinate multiple filter: a sequence of skippedinput spans by
iteratively projected patterns, as described in imagination section of part 3.
Alternatively, negative coordinate filters implement crosslevel shortcuts,
described in level 3 subpart, which select for projected matchassociated
novelty.
Additional variables in
positive patterns increase cost, which decreases positive vs. negative span
proportion.
Increased difference in
sign, syntax, span, etc., also reduces match between positive and negative
patterns. So, comparison, evaluation, prevaluation... on
higher levels is primarily for samesign patterns.
Consecutive differentsign
patterns are compared due to their proximity, forming ratios of their span and
other variables. These ratios are applied to project match across
differentsign gap or contrast pattern:
projected match +=
(projected match  intervening negative match) * (negative value / positive
value) / 2?
Î›V selection is incremented
by induction: forward and feedback of actual inputs, or by deduction: algebraic compression of input syntax, to find computational
shortcuts. Deduction is faster, but actual inputs also
carry empirical information. Relative value of additive information vs.
computational shortcuts is set by feedback.
Following subparts cover
three initial levels of search in more detail, though out of date:
Level 1: comparison to past
inputs, forming difference patterns and match patterns
Inputs to the 1^{st} level of search are single integers, representing pixels of 1D
scan line across an image, or equivalents from other modalities. Consecutive inputs are compared to form differences, difference
patterns, matches, relative match patterns. This comparison may be extended,
forming higher and distant derivatives:
resulting variables per
input: *=2 derivatives (d,m) per comp, + conditional *=2 (xd, xi) per extended
comp:
8 derivatives
// ddd, mdd, dd_i, md_i, + 1inputdistant dxd,
mxd, + 2inputdistant d_ii, m_ii,
/
\
4 der
4 der // 2 consecutive: dd, md, + 2
derivatives between 1inputdistant inputs: d_i and m_i,
/
\ / \
d,m d,m
d,m // d, m: derivatives from default comparison between consecutive
inputs,
/ \ /
\ / \
i >> i
>> i >> i // i: singlevariable inputs.
This is explained /
implemented in my draft python code: level_1_working. That first level is for generic 1D cognitive algorithm, its
adaptation for image and then video recognition algorithm will be natively 2D.
That’s what I spend most
of my time on, the rest of this intro is significantly out of date.
bitfiltering and
digitization
1^{st} level inputs are filtered by the value of most and least
significant bits: maximal and minimal detectable magnitude of inputs. Maximum
is a magnitude that cooccurs with average 1^{st} level
match, projected by outputted dP s. Least significant bit value is determined by maximal value and
number of bits per variable.
This bit filter is
initially adjusted by overflow in 1^{st} level
inputs, or by a set number of consecutive overflows.
It’s also adjusted by
feedback of higherlevel patterns, if they project over or under flow of 1^{st} level inputs that exceeds the cost of adjustment. Underflow is
average number of 0 bits above top 1 bit.
Original input resolution
may be increased by projecting analog magnification, by impact or by distance.
Iterative bitfiltering is
digitization: bit is doubled per higher digit, and exceeding summed input is
transferred to next digit. A digit can be larger than binary if the cost of
such filtering requires larger carry.
Digitization is the most
basic way of compressing inputs, followed by comparison between resulting
integers.
hypothetical: comparable
magnitude filter, to form minimalmagnitude patterns
Initial magnitude justifies
basic comparison, and summation of belowaverage inputs only compensates for
their lower magnitude, not for the cost of conversion. Conversion involves
higherpower comparison, which must be justified by higher order of match, to
be discovered on higher levels. Or by average neg. mag. span?
iP min mag span conversion cost and comparison
match would be on 2^{nd} level, but it’s not justified by 1^{st} level match, unlike D span conversion cost and comparison match,
so it is effectively the 1^{st} level of comparison?
possible +iP span evaluation: double evaluation + span representation cost
< additional lowerbits match?
The inputs may be
normalized by subtracting feedback of average magnitude, forming ± deviation,
then by dividing it by next+1 level feedback, forming a multiple of average
absolute deviation, and so on. Additive value of input is a combination of all
deviation orders, starting with 0^{th} or
absolute magnitude.
Initial input evaluation if
any filter: cost < gain: projected negativevalue (comparison cost  positive
value):
by minimal magnitude > ± relative magnitude patterns (iP s), and + iP s are evaluated or crosscompared?
or by average magnitude
> ± deviations, then by coaverage
deviation: ultimate bit filter?
Summation *may* compensate for conversion if its span is greater than average per
magnitude spectrum?!
Summation on higher levels
also increases span order, but withinorder conversion is the same, and
betweenorder comparison is intrapattern only. bf spans overlap vP span, > filter conversion costs?
Level 2: additional evaluation of input patterns for feedback, forming
filter patterns (out of date)
Inputs to 2^{nd} level of search are patterns derived on 1^{st} level: dP ( L, I, D, V, Q (d)) s or t dP ( tV, tD, dP ()) s.
These inputs are evaluated
for feedback to update 0^{th} level i LSB, terminating samefilter span.
Feedback increment of LSB is evaluated by deviation (∆) of magnitude, to avoid input overflow or
underflow:
∆ += I/ L  LSB a; ∆ > ff? while (∆ > LSB a){ LSB ±; ∆ = LSB a; LSB a *2};
LSB a is average input (* V/ L?) per LSB value, and ff is average deviation per
positivevalue increment;
Î£ (∆) before evaluation: no V patterns? #b++ and C LSB are more expensive,
evaluated on 3^{rd} level?
They are also compared to
previously inputted patterns, forming difference patterns dPs and value patterns vPs per input variable, then combined into dPP s and vPP s per input pattern.
L * sign of consecutive dP s is a known miss, and
match of dP variables is correlated by common derivation.
Hence, projected match of
other +dP and dP variables = amk * (1  L / dP). On the other hand, samesign dP s are distant by L, reducing projected match by amk * L, which is equal to reduction by miss of L?
So, dP evaluation is for two
comparisons of equal value: crosssign, then cross L samesign (1 dP evaluation is blocked by feedback of discovered or defined alternating
sign and covariable match projection).
Both of last dP s will be compared to the next one, thus past match per dP (dP M) is summed for three dP s:
dP M ( Î£ ( last 3 dP s L+M))  a dP M (average of 4Le +vP dP M) > v, vs;; evaluation / 3 dP s > value, sign / 1 dP.
while (vs = ovs){ ovs = vs; V+=v; vL++; vP (L, I, M, D) += dP (L, I, M, D);; default vP  wide sum, select preserv.
vs > 0? comp (3 dP s){ DIV (L, I, M, D) > N, ( n, f, m, d); vP (N, F, M, D) += n, f, m, d;; sum: der / variable, n / input?
vr = v+ N? SUB (nf) > nf m; vd = vr+ nf m, vds = vd  a;; ratios are too small for DIV?
while (vds = ovds){ ovds = vds; Vd+=vd; vdL++; vdP() += Q (d  ddP);; default Q (d  ddP) sum., select. preserv.
vds > 0? comp (1^{st} x l^{st} d  ddP s of Q (d) s);; splicing Q (d) s of matching dP s, cont. only: no comp ( Î£ Q (d  ddP)?
Î£ vP ( Î£ vd P eval: primary for P, redundant to individual dP s ( d s for +P, cost *2, same for
+P' I and P' M,D?
no Î£ V  Vd evaluation of cont. comp per variable or division: cost + vL = comp cost? Î£ V per fb: no vL, #comp;
 L, I, M, D: same value per mag, power / compression, but I  M, D redund = mag, +vP: I  2a,  vP: M, D  2a?
 no
variable eval: cost (sub + vL + filter) > comp cost, but match value must be
adjusted for redundancy?
 normalization
for comparison: min (I, M, D) * rL, SUB (I, M, D)? Î£ L (pat) vs C: more general but
interrupted?
variablelength DIV: while (i > a){ while (i> m){ SUB (i, m) > d; n++; i=d;}; m/=2; t=m; SUB (d, t); f+= d;}?
additive compression per d vs. m*d: > length cost?
tdP ( tM, tD, dP(), ddP Î£ ( dMÎ£ (Q (dM)), dDÎ£ (Q (dD)), ddLÎ£ (Q (ddL)), Q (ddP))); // last d
and D are within dP()?
Input filter is a
higherlevel average, while filter update is accumulated over multiple
higherlevel spans until it exceeds filterupdate filter. So, filter update is
2^{nd} order feedback relative to filter, as is filter
relative to match.
But the same filter update
is 3^{rd} order of feedback when used to evaluate
input value for inclusion into pattern defined by a previous filter: update
span is two orders higher than value span.
Higherlevel comparison
between patterns formed by different filters is mediated, vs. immediate
continuation of currentlevel comparison across filter update (mediated cont.:
splicing between differentfilter patterns by vertical specification of match,
although it includes lateral crosscomparison of skipdistant specifications).
However, filter update
feedback is periodic, so it doesn’t form continuous crossfilter comparison
patterns xPs.
adjustment of forward
evaluation by optional feedback of projected input
More precisely, additive
value or novel magnitude of an input is its deviation from higherlevel
average. Deviation = input  expectation: (higherlevel summed input 
summed difference /2) * rL (L / hL).
Inputs are compared to last
input to form difference, and to past average to form deviation or novelty.
But last input is more
predictive of the next one than a more distant average, thus the latter is
compared on higher level than the former. So, input variable is compared
sequentially and summed within resulting patterns. On the next level, the sum
is compared vertically: to nextnextlevel average of the same variable.
Resulting vertical match
defines novel value for higherlevel sequential comparison:
novel value = past match  (vertical
match * higherlevel match rate)  average novel match:
nv = L+M  (m (I, (hI * rL)) * hM / hL)  hnM * rL; more precise than initial value: v = L+M  hM * rL;
Novelty evaluation is done
if higherlevel match > cost of feedback and operations,
separately for I and D P s:
I, M ( D, M feedback, vertical SUB (I, nM ( D, ndM));
Impact on ambient sensor is
separate from novelty and is predicted by representationalvalue patterns?
 nextinput
prediction: seq match + vert match * relative rate, but
predictive selection is per level, not input.
 higherorder
expectation is relative match per variable: pMd = D * rM, M/D, or D * rMd: Md/D,
 if rM 
rMd are derived by intrapattern comparison, when average M  Md >
average per division?
oneinput search extension
within crosscompared patterns
Match decreases with
distance, so initial comparison is between consecutive inputs. Resulting match
is evaluated, forming ±vP s. Positive P s are then evaluated for expanded internal search:
crosscomparison among 1inputdistant inputs within a pattern (on same level,
higherlevel search is between new patterns).
This cycle repeats to
evaluate crosscomparison among 2inputdistant inputs, 3inputdistant inputs,
etc., when summed currentdistance match exceeds the average per evaluation.
So, patterns of longer
crosscomparison range are nested within selected positive patterns of shorter
range. This is similar to 1^{st} level ddP s being nested within dP s.
Same input is reevaluated
for comparison at increased distance because match will decay: projected match
= last match * match rate (mr), * (higherlevel
mr / currentlevel mr) * (higherlevel distance /
next distance)?
Or = input * average match rate for that specific
distance, including projected match within negative patterns.
It is reevaluated also
because projected match is adjusted by past match: mr *= past mr / past projected mr?
Also, multiple comparisons
per input form overlapping and redundant patterns (similar to fuzzy clusters),
and must be evaluated vs.
filter * number of prior comparisons, reducing value of projected match.
Instead of directly
comparing incrementally distant input pairs, we can calculate their difference
by adding intermediate differences. This would obviate multiple access to the
same inputs during crosscomparison.
These differences are also
subtracted (compared), forming higher derivatives and matches:
ddd, x1dd, x2d ( ddd: 3^{rd}
derivative, x1dd: d of 2inputdistant d s, x2d: d of 2inputdistant
inputs)
/ \
dd, x1d dd, x1d ( dd: 2^{nd} derivative, x1d = d+d = difference between 1inputdistant
inputs)
/
\
/ \
d d d ( d: difference between consecutive inputs)
/ \ /
\ / \
i
i
i i
( i: initial inputs)
As always, match is a
smaller input, cached or restored, selected by the sign of a difference.
Comparison of both types is
between all sametype variable pairs from different inputs.
Total match includes match
of all its derivation orders, which will overlap for proximate inputs.
Incremental cost of
crosscomparison is the same for all derivation orders. If projected match is
equal to projected miss, then additive value for different orders of the same
inputs is also the same: reduction in projected magnitude of differences will
be equal to reduction in projected match between distant inputs?
multiinput search
extension, evaluation of selection per input: tentative
On the next level, average
match from expansion is compared to that from shorterdistance comparison, and
resulting difference is decay of average match with distance. Again,
this decay drives reevaluation per expansion: selection of inputs with
projected decayed match above average per comparison cost.
Projected match is also
adjusted by prior match (if local decay?) and redundancy (symmetrical
if no decay?)
Slower decay will reduce
value of selection per expansion because fewer positive inputs will turn
negative:
Value of selection = Î£ comp cost of negvalue inputs  selection cost (average saved cost or relative delay?)
This value is summed
between higherlevel inputs, into average value of selection per increment of
distance. Increments with negative value of selection should be compared
without reevaluation, adding to minimal number of comparisons per selection,
which is evaluated for feedback as a comparisondepth filter:
Î£ (selection value per
increment) > average selection value;; for negative patterns of each depth,  >1 only?
depth adjustment value =
average selection value; while (average selection value > selection cost){
depth adjustment ±±; depth adjustment value = selection value per
increment (depthspecific?); };
depth adjustment >
minimal per feedback? >> lowerlevel depth filter;; additive depth = adjustment value?
 match filter is summed
and evaluated per current comparison depth?
 selected positive relative
matches don’t reduce the benefit of pruningout negative ones.
 skip if negative
selection value: selected positive matches < selection
cost: average value or relative delay?
Each input forms a queue of
matches and misses relative to templates within comparison depth filter. These
derivatives, both discrete and summed, overlap for inputs within each other’s
search span. But representations of discrete derivatives can be reused,
redundancy is only necessary for parallel comparison.
Assuming that environment
is not random, similarity between inputs declines with spatiotemporal
distance. To maintain proximity, a ninput search is FIFO: input is compared to
all templates up to maximal distance, then added to the queue as a new
template, while the oldest template is outputted into patternwide queue.
valueproportional
combination of patterns:
tentative
Summation of +dP and dP is
weighted by their value: L (summed dsign match) + M (summed i match).
Such relative probability
of +dP vs.  dP is indicated by corresponding ratios: rL =
+L/L, and rM = +M/M.
(Ls and Ms are compared by
division: comparison power should be higher for more predictive variables).
But weighting
complementation incurs costs, which must be justified by value of ratio. So,
division should be of variable length, continued while the ratio is above
average. This is shown below for Ls, also applies to Ms:
dL = +L  L, mL = min (+L, L); nL =0; fL=0; efL=1; // nL: L multiple, fL: L fraction,
efL: extended fraction.
while (dL > adL){ dL =
dL; // all Ls are positive; dL is evaluated for long division by adL: average
dL.
while (dL > 0){ dL =
mL; nL++;} dL = mL/2; dL >0? fL+= efL; efL/=2;} // ratio: rL= nL + fL.
Ms’ longdivision
evaluation is weighted by rL: projected rM value = dM * nL (reducedresolution rL)  adM.
Ms are then combined: cM =
+M + M * rL; // rL is relative probability of M across iterated cL.
Ms are not projected (M+= D * rcL * rM D (MD/cD) /2): precision of higherlevel rM D is below that of rM?
Prior ratios are
combination rates: rL is probability of M, and combined rL and rM (cr) is probability of D.
If rM < arM, cr = rL,
else: cr = (+L + +M) / (L + M) // cr = √(rL * rM) would lose L vs. M
weighting.
cr predicts match of
weighted cD between cdPs, where negativedP variable is multiplied by
aboveaverage match ratio before combination: cD = +D + D * cr. // after unweighted comparison between Ds?
Averages: arL, arM, acr, are feedback of ratios that cooccur with aboveaverage match of
spannormalized variables, vs. input variables. Another feedback is averages that evaluate long division: adL, adM, adD.
Both are feedback of
positive C pattern, which represents these variables, inputted & evaluated on 3^{rd} level.
; or 4^{th} level: value of dPs * ratio is compared to value of dPs, & the difference is multiplied by cL / hLe cL?
Comparison of oppositesign
Ds forms negative match = smaller D, and positive difference dD = +D+ D.
dD magnitude predicts its
match, not further combination. Single comparison is cheaper than its evaluation.
Comparison is by division
if larger D cooccurs with hLe nD of aboveaverage predictive value (division
is signneutral & reductive). But average nD value is below the cost of
evaluation, except if positive feedback?
So, default operations for
L, M, D of complementary dPs are comparison by long division and combination.
D combination: +D D*cr, vs. 
cD * cr: +D vs. D weighting is lost, meaningless if
cD=0?
Combination by division is
predictive if the ratio is matching on higher level (hLe) & acr is fed back as filter?
Resulting variables: cL,
rL, cM, rM, cr, cD, dD, form top level of cdP: complemented dP.
Level 3: prevaluation of
projected filter patterns, forming updatedinput patterns
(out of date)
3^{rd} level inputs are ± V patterns, combined into
complemented V patterns. Positive V patterns include derivatives of 1^{st} level match, which project match within future inputs (D patterns only represent and project derivatives of magnitude).
Such projectedinputsmatch is prevaluated, negative prevaluespan inputs are summed or
skipped (reloaded), and positive prevaluespan inputs are evaluated or even
directly compared.
Initial upward (Î›) prevaluation by E filter selects for evaluation of V patterns, within resulting
± E patterns. Resulting prevalue is also projected
downward (V), to select future input spans for evaluation, vs. summation or
skipping. The span is of projecting V pattern, same as of lower
hierarchy. Prevaluation is then iterated over multiple projectedinput spans,
as long as last prevalue remains above average for the cost of prevaluation.
Additional interference of
iterated negative projection is stronger than positive projection of lower
levels, and should flush them out of pipeline. This flushing need not be final,
spans of negative projected value may be stored in buffers, to delay the loss.
Buffers are implemented in slower and cheaper media (tape vs. RAM) and accessed
if associated patterns match on a higher level, thus project aboveaverage
match among their inputs.
Iterative backprojection
is evaluated starting from 3^{rd} level:
to be projectable the input must represent derivatives of value, which are
formed starting from 2^{nd} level. Compare this to 2^{nd} level evaluation:
Î› for input, V for V filter, iterated within V pattern. Similar
subiteration in E pattern?
Evaluation value =
projectedinputsmatch  E
filter: average input match that
cooccurs with average higherlevel match per evaluation (thus accounting
for evaluation costs + selected comparison costs). Compare this to V filter that selects for 2^{nd} level
comparison: average input match that cooccurs with average higherlevel match
per comparison (thus accounting for costs of default crosscomparison only).
E filter feedback starts from 4^{th} level of search, because its inputs represent prevaluated
lowerlevel inputs.
4^{th} level also preprevaluates vs. prevaluation filter, forming preprevalue
that determines prevaluation vs. summation of next input span. And so on: the
order of evaluation increases with the level of search.
Higher levels are
increasingly selective in their inputs, because they additionally select by
higher orders derived on these levels: magnitude ) match and difference of
magnitude ) match and difference of match, etc.
Feedback of prevaluation is
± prefilter: binary evaluationvalue sign that determines evaluating vs.
skipping initial inputs within projected span, and flushing those already
pipelined within lower levels.
Negative feedback may be
iterated, forming a skip span.
Parallel lower hierarchies & skip spans may be assigned to different external sources or their
internal buffers.
Filter update feedback is
levelsequential, but prefilter feedback is sent to all lower levels at once.
Prefilter is defined per
input, and then sequentially translated into prefilters of higher derivation
levels:
prior value += prior match
> value sign: nextlevel prefilter. If there are multiple prefilters of
different evaluation orders from corresponding levels, they AND & define
infrapatterns: sign ( input ( derivatives.
filter update evaluation and feedback
Negative evaluationvalue blocks input evaluation (thus
comparison) and filter updating on all lower levels. Notevaluated input spans (gaps)
are also outputted, which will increase coordinate range per contents of both
higherlevel inputs and lowerlevel feedback. Gaps represent negative projectedmatch value, which must be combined with positive value of subsequent
span to evaluate comparison across the gap on a higher level. This is similar
to evaluation of combined positive + negative relative match spans, explained
above.
Blocking locations with
expected inputs will result in preference for exploration & discovery of new patterns, vs. confirmation of the old ones. It
is the opposite of upward selection for stronger patterns, but sign reversal in
selection criteria is basic feature of any feedback, starting with average
match & derivatives.
Positive evaluationvalue input spans are evaluated by
lowerlevel filter, & this filter is evaluated for update:
combined update = (output
update + output filter update / (samefilter span (fL) / output span)) /2.
both updates: = last
feedback, equalweighted because higherlevel distance is compensated by range:
fL?
update value = combined
update  update filter: average update per average higherlevel additive
match.
also differential costs of
feedback transfer across locations (vs. delay) + representation +
filter conversion?
If update value is negative: fL += new inputs, subdivided by their positive or
negative predictive value spans.
If update value is positive: lowerlevel filter += combined update, new fL (with new filter representation) is initialized on a current
level, while currentlevel part of old fL is outputted and evaluated
as nextlevel input.
In turn, the filter gets
updates from higherlevel outputs, included in higherhigherlevel positive
patterns by that level’s filter. Hence, each filter represents combined
spannormalized feedback from all higher levels, of exponentially growing span
and reduced update frequency.
Deeper hierarchy should
block greater proportion of inputs. At the same time, increasing number of
levels contribute to projected additive match, which may justify deeper search
within selected spans.
Higherlevel outputs are
more distant from current input due to elevation delay, but their projection
range is also greater. So, outputs of all levels have the same relative distance (distance/range) to a next input, and are equalweighted in combined update. But if input span is skipped, relative
distance of skipinitiating pattern to next input span will increase, and its
predictive value will decrease. Hence, that pattern should be flushed or at
least combined with a higherlevel one:
combined V prevalue = higherlevel V prevalue + ((currentlevel V prevalue  higherlevel V prevalue) / ((currentlevel span / distance) / (higherlevel span / distance)) /2. // the difference between
currentlevel and higherlevel prevalues is reduced by the ratio of their
relative distances.
To speed up selection,
filter updates can be sent to all lower levels in parallel. Multiple direct
filter updates are spannormalized and compared at a target level, and the
differences are summed in combined update. This combination is equalweighted
because all levels have the same spanperdistance to next input, where the
distance is the delay of feedback during elevation. // this happens automatically
in levelsequential feedback?
combined update = filter update + distancenormalized difference between output
& filter updates:
((output update  filter
update) / (output relative distance / higheroutput
relative distance)) /2.
This combination method is
accurate for postskipped input spans, as well as next input span.
 filter can also be
replaced by output + higherlevel filter /2, but value of such feedback is not known.
 possible fixedrate
sampling, to save on feedback evaluation if slow decay, ~ deep feedforward
search?
 selection can be by
patterns, derivation orders, subpatterns within an order, or individual
variables?
 match across distance
also projects across distance: additive match = relative match * skipped
distance?
crosslevel shortcuts: higherlevel subfilters and symbols
After individual input
comparison, if match of a current scale (lengthofalength…) projects positive
relative match of input lowerscale / higherderivation level, then the later
is also crosscompared between the inputs.
Lower scale levels of a
pattern represent old lower levels of a search hierarchy (current or buffered
inputs).
So, feedback of lower scale
levels goes down to corresponding search levels, forming shortcuts to preserve detail for higher levels. Feedback is generally negative:
expectations are redundant to inputs. But specifying feedback may be positive:
lowerlevel details are novel to a pattern, &
projected to match with it in the future.
Higherspan comparison
power is increased if lowerspan comparison match is below average:
variable subtraction ) span division )
superspan logarithm?
Shortcuts to individual
higherlevel inputs form a queue of subfilters on a lower level, possibly
represented by a queuewide prefilter. So, a level has one filter per parallel
higher level, and subfilter for each specified subpattern.
Subfilters of incrementally distant inputs are redundant to all previous ones.
Corresponding input value = match  subfilter value * rate of match to subfilter
* redundancy?
Shortcut to a whole level
won’t speedup search: higherlevel search delay > lowerhierarchy search
delay.
Resolution and parameter
range may also increase through interaction of colocated counterprojections?
Symbols, for communication
among systems that have common highlevel concepts but no direct interface, are
“coauthor identification” shortcuts: their recognition and interpretation is
performed on different levels.
Higherlevel patterns have
increasing number of derivation levels, that represent corresponding lower
search levels, and project across multiple higher search levels, each evaluated
separately?
Match across discontinuity may be due to additional
dimensions or internal gaps within patterns.
Search depth may also be
increased by crosscomparison between levels of scale within a
pattern: match across multiple scale levels also projects over multiple higher
and lower scale levels? Such comparison between variable types within a
pattern would be of a higher order:
5. Comparison between
variable types within a pattern (tentative)
To reiterate, elevation
increases syntactic complexity of patterns: the number of different variable
types within them. Syntax is identification of these types by their position
(syntactic coordinate) within a pattern. This is analogous to recognizing parts
of speech by their position within a sentence.
Syntax “synchronizes”
sametype variables for comparison  aggregation between input patterns. Access
is hierarchical, starting from sign>value levels within each variable of difference
and relative match: sign is compared first, forming + and  segments, which are
then evaluated for comparison of their values.
Syntactic expansion is
pruned by selective comparison vs. aggregation of individual variable types
within input patterns, over each coordinate type or resolution. As with
templates, minimal aggregation span is resolution of individual inputs, &
maximal span is determined by average magnitude (thus match) of new derivatives
on a higher level. Hence, a basic comparison cycle generates queues of
interlaced individual & aggregate derivatives at each template variable,
and conditional higher derivatives on each of the former.
Sufficiently complex syntax
or predictive variables will justify comparing across “syntactic“ coordinates
within a pattern, analogous to comparison across external coordinates. In fact,
that’s what higherpower comparisons do. For example, division is an iterative
comparison between difference & match: within a pattern (external
coordinate), but across derivation (syntactic coordinate).
Also crossvariable is
comparison between orders of match in a pattern: magnitude, match,
matchofmatch... This starts from comparison between match & magnitude:
match rate (mr) = match / magnitude. Match
rate can then be used to project match from magnitude: match = magnitude * output mr * filter mr.
In this manner, mr of each match order adjusts intraorderderived sequentially
higherorder match:
match *= lower interorder mr. Additive match is then projected from adjusted matches &
their derivatives.
This interorder projection
continues up to the top order of match within a pattern, which is the ultimate
selection criterion because that’s what’s left matching on the top level of
search.
Interorder vectors are Î›V symmetrical, but Î›V derivatives from lower order of match are also
projected for higherorder match, at the same rate as the match itself?
Also possible is comparison
across syntactic gaps: Î›Y comparison > difference, filter feedback VY hierarchy. For example, comparison between dimensions of a
multiD pattern will form possibly recurrent proportions.
Internal comparisons can
further compress a pattern, but at the cost of adding a higherorder syntax,
which means that they must be increasingly selective. This selection will
increase “discontinuity” over syntactic coordinates: operations necessary to
convert the variables before comparison. Eventually, such operators will become
large enough to merit direct comparisons among them. This will produce algebraic
equations, where the match (compression) is a reduction in the number of
operations needed to produce a result.
The first such shortcut
would be a version of Pythagorean theorem, discovered during search in 2D (part
6) to compute cosines. If we compare 2Dadjacent 1D Ls by division, over 1D
distance and derivatives (an angle), partly matching ratio between the ratio of
1D Ls and a 2nd derivative of 1D distance will be a cosine.
Cosines are necessary to
normalize all derivatives and lengths (Ls) to a value they have when orthogonal
to 1D scan lines (more in part 6).
Such normalization for a
POV angle is similar to dimensionality reduction in Machine Learning, but is much more efficient
because it is secondary to selective dimensionality expansion. It’s not really
“reduction”: dimensionality is prioritized rather than reduced. That is, the
dimension of pattern’s main axis is maximized, and dimensions sequentially
orthogonal to higher axes are correspondingly minimized. The process of
discovering these axes is so basic that it might be hardwired in animals.
6. Cartesian dimensions and
sensory modalities (out of date)
This is a recapitulation
and expansion on incremental dimensionality introduced in part 2.
Term “dimension” here is
reserved for a parameter that defines sequence and distance among inputs,
initially Cartesian dimensions + Time. This is different from terminology of
combinatorial search, where dimension is any parameter of an input, and their
external order and distance don’t matter. My term for that is “variable“,
external dimensions become types of a variable only after being encoded within
input patterns.
For those with ANN
background, I want to stress that a level of search in my approach is 1D queue
of inputs, not a layer of nodes. The inputs to a node are combined regardless
of difference and distance between them (the distance is the difference between
laminar coordinates of source “neurons”).
These derivatives are
essential because value of any prediction = precision of what * precision of where. Coordinates and coderived differences are not
represented in ANNs, so they can't be used to calculate Euclidean vectors.
Without such vectors, prediction and selection of where must remain extremely crude.
Also, layers in ANN are
orthogonal to the direction of input flow, so hierarchy is at least 2D. The
direction of inputs to my queues is in the same dimension as the queue itself,
which means that my core algorithm is 1D. A hierarchy of 1D queues is the most
incremental way to expand search: we can add or extend only one coordinate at a
time. This allows algorithm to select inputs that are predictive enough to
justify the cost of representing additional coordinate and corresponding
derivatives. Again, such incremental syntax expansion is my core principle,
because it enables selective (thus scalable) search.
A common objection is that
images are “naturally” 2D, and our spacetime is 4D. Of course, these empirical
facts are practically universal in our environment. But, a core cognitive
algorithm must be able to discover and forget any empirical specifics on its
own. Additional dimensions can be discovered as some general periodicity in the
input flow: distances between matching inputs are compared, match between these
distances indicates a period of lower dimension, and recurring periods form
higherdimension coordinate.
But as a practical shortcut
to expensive dimensiondiscovery process, initial levels should be designed to
specialize in sequentially higher spatial dimensions: 1D scan lines, 2D frames,
3D set of confocal “eyes“, 4D temporal sequence. These levels discover
contiguous (positive match) patterns of increasing dimensionality:
1D line segments, 2D blobs,
3D objects, 4D processes. Higher 4D cycles form hierarchy of multidimensional
orders of scale, integrated over time or distributed sensors. These higher
cycles compare discontinuous patterns. Corresponding dimensions may not be
aligned across cycles of different scale order.
Explicit coordinates and
incremental dimensionality are unconventional. But the key for scalable search
is input selection, which must be guided by costbenefit analysis. Benefit is projected
match of patterns, and cost is representational complexity per pattern. Any
increase in complexity must be justified by corresponding increase in
discovered and projected match of selected patterns. Initial inputs have no
known match, thus must have minimal complexity: singlevariable “what”, such as
brightness of a greyscale pixel, and singlevariable “where”: pixel’s
coordinate in one Cartesian dimension.
Single coordinate means
that comparison between pixels must be contained within 1D (horizontal) scan
line, otherwise their coordinates are not comparable and can’t be used to
select locations for extended search. Selection for contiguous or proximate
search across scan lines requires second (vertical) coordinate. That increases
costs, thus must be selective according to projected match, discovered by past
comparisons within 1D scan line. So, comparison across scan lines must be done
on 2^{nd} level of search. And so on.
Dimensions are added in the
order of decreasing rate of change. This means spatial dimensions are scanned
first: their rate of change can be spedup by moving sensors. Comparison over
purely temporal sequence is delayed until accumulated change / variation
justifies search for additional patterns. Temporal sequence is the original
dimension, but it is mapped on spatial dimensions until spatial continuum is
exhausted. Dimensionality represented by patterns is increasing on higher
levels, but each level is 1D queue of patterns.
Also independently
discoverable are derived coordinates: any variable with cumulative match that correlates
with combined cumulative match of all other variables in a pattern. Such
correlation makes a variable useful for sequencing patterns before
crosscomparison.
It is discovered by summing
matches for sametype variables between input patterns, then crosscomparing
summed matches between all variables of a pattern. Variable with the highest
resulting match of match (mm) is a candidate coordinate. That mm is then
compared to mm of current coordinate. If the difference is greater than cost of
reordering future inputs, sequencing feedback is sent to lower levels or
sensors.
Another type of empirically
distinct variables is different sensory modalities: colors, sound and pitch,
and so on, including artificial senses. Each modality is processed separately,
up a level where match between patterns of different modalities but same scope
exceeds match between unimodal patterns across increased distance. Subsequent
search will form multimodal patterns within common ST frame of reference.
As with external dimensions,
difference between modalities can be predefined or discovered. If the latter,
inputs of different modalities are initially mixed, then segregated by
feedback. Also as with dimensions, my core algorithm only assumes singlemodal
inputs, predefining multiple modalities would be an addon.
7. Notes on clustering,
ANNs, and probabilistic inference (out of date)
In terms of conventional
machine learning, my approach is a form of hierarchical fuzzy clustering. Cluster is simply a different term for
pattern: a set of matching inputs. Each set is represented by a centroid: an
input with belowthreshold combined “distance” to other inputs of the set. The
equivalent of centroid in my model is an input with aboveaverage match (a
complementary of a distance) to other inputs within its search span. Such
inputs are selected to search nextlevel queue and to indirectly represent other
crosscompared but not selected inputs, via their discrete or aggregate
derivatives relative to the selected one.
Crucial differences here is
that conventional clustering methods initialize centroids with arbitrary random
weights, while I use matches (and so on) from past comparisons. And the weights
are usually defined in terms of one variable, while I select higherlevel
inputs based on a combination of all variables per pattern, the number of which
increases with the level of search.
Current methods in
unsupervised learning were developed / accepted because they solved specific
problems with reasonable resources. But, they aren’t comparable to human
learning in scalability. I believe that requires an upfront investment in
incremental complexity of representation: a syntactic overhead that makes such
representation uncompetitive at short runs, but is necessary to predictively
prune longerrange search.
The most basic example here
is my use of explicit coordinates, and of input differences at their distances.
I haven’t seen that in other lowlevel approaches, yet they are absolutely
necessary to form Euclidean vectors. Explicit coordinates is why I start image
processing with 1D scan lines,  another thing that no one else does. Images
seem to be “naturally” 2D, but expanding search in 2Ds at once adds the extra
cost of two sets of new coordinates & derivatives. On the other hand,
adding 1D at a time allows to select inputs for each additional layer of
syntax, reducing overall (number of variables * number of inputs) costs of
search on the next level.
Artificial Neural Networks
The same coordinateblind
mindset pervades ANNs. Their learning is probabilistic: match is determined as
an aggregate of multiple weighted inputs. Again, no derivatives (0D miss), or
coordinates (14D miss) per individual input pair are recorded. Without them,
there is no Euclidean vectors, thus pattern prediction must remain extremely
crude. I use aggregation extensively, but this degradation of resolution is
conditional on results of prior comparison between inputs. Neurons simply don’t
have the capacity for primary comparison.
This creates a crucial
difference in the way patterns are represented in the brain vs. my hierarchy of
queues. Brain consists of neurons, each likely representing a variable of a
given magnitude and modality. These variables are shared among multiple
patterns, or coactivated networks (“cognits” in terms of J. Fuster).
This is conceptually
perverse: relatively constant values of specific variables define a pattern,
but there’s no reason that same values should be shared among different
patterns. The brain is forced to share variables because it has fixed number of
neurons, but a fluid and far greater number of their networks.
I think this is responsible
for our crude taxonomy, such as using large, medium, small instead of specific
numbers. So, it’s not surprising that our minds are largely irrational, even
leaving aside all the subcortical nonsense. We don’t have to slavishly copy
these constraints. Logically, a cooccurring set of variables should be
localized within a pattern. This requires more local memory for redundant
representations, but will reduce the need for interconnect and transfers to
access global shared memory, which is far more expensive.
More broadly, neural
networkcentric mindset itself is detrimental, any function must be initially
conceptualized as a sequential algorithm, parallelization is an optional
superstructure.
Probabilistic inference: AIT,
Bayesian logic, Markov models
A good introduction to
Algorithmic information theory is Philosophical Treatise of Universal Induction by Rathmanner and Hutter. The criterion is same
as mine: compression and prediction. But, while a progress vs. frequentist
probability calculus, both AIT and Bayesian inference still assume a prior,
which doesn’t belong in a consistently inductive approach. In my approach,
priors or models are simply past inputs and their patterns. Subjectspecific
priors could speedup learning, but unsupervised pattern discovery algorithm
must be the core on which such shortcuts are added or removed from.
More importantly, as with
most contemporary approaches, Bayesian learning is statistical and
probabilistic. Probability is estimated from simple incidence of events, which
I think is way too coarse. These events hardly ever match or miss precisely, so
their similarity should be quantified. This would add a whole new dimension of
micrograyscale: partial match, as in my approach, vs. binary incidence in
probabilistic inference. It should improve accuracy to the same extent that
probabilistic inference improved on classical logic, by adding a
macrograyscale of partial probability vs. binary true  false values of the
former.
Resolution of my inputs is
always greater than that of their coordinates, while Bayesian inference and AIT
typically start with the reverse: strings of 1bit inputs. These inputs, binary
confirmations / disconfirmations, are extremely crude way to represent “events”
or inputs. Besides, the events are assumed to be highlevel concepts: the kind
that occupy our conscious minds and are derived from senses by subconscious
cognitive processes, which must be built into general algorithm. Such choice of
initial inputs in BI and AIT demonstrates a lack of discipline in incrementing
complexity.
To attempt a general
intelligence, Solomonoff introduced “universal prior“: a class of all models.
That class is a priori infinite, which means that he hits combinatorial
explosion even *before* receiving actual inputs. It‘s a solution that only a
mathematician may find interesting. Marginally practical implementation of AIT
is Levin Search, which randomly generates models / algorithms of incremental
complexity and selects those that happen to solve a problem or compress a bit
string.
Again, I think starting
with prior models is putting a cart before a horse: cognition must start with
raw data, complex math only becomes costefficient on much higher levels of
selection and generalization. And this distinction between input patterns and
pattern discovery process is only valid within a level: algorithm is embedded
in resulting patterns, hence is also compared on higher levels, forming
“algorithmic patterns“.
8. Notes on working mindset
and a prize for contributions
My terminology is as general
as the subject itself. It’s a major confounder,  people crave context, but
generalization is decontextualization. And cognitive algorithm is a
metageneralization: the only thing in common for everything we learn. This
introduction is very compressed, because much of it is work in progress. But I
think it also reflects and cultivates ruthlessly reductionist mindset required
for such subject.
My math is very simple,
because algorithmic complexity must be incremental. Advanced math can
accelerate learning on higher levels of generalization, but it’s too expensive
for initial levels. And minimal general learning algorithm must be able to
discover computational shortcuts (AKA math) on it’s own, just like we do.
Complex math is definitely not innate in humans on any level: cavemen didn’t do
calculus.
This theory may seem too speculative,
but any degree of generalization must be correspondingly lossy. Which is
contrary to precisionoriented culture of math and computer science. Hence,
current Machine Learning is mostly experimental, and the progress on
algorithmic side is glacial. A handful of people aspire to work on AGI, but they either lack or neglect functional definition of
intelligence, their theories are only vague inspiration.
I think working on this
level demands greater delay of experimental verification than is acceptable in
any established field. Except for philosophy, which has nothing else real to
study. But established philosophers have always been dysfunctional fluffers,
not surprisingly as their only paying customers are college freshmen.
Our main challenge in
formalizing GI is a speciewide ADHD. We didn’t evolve for sustained focus on
this level of generalization, that would cause extinction long before any
tangible results. Which is no longer a risk, GI is the most important problem
conceivable, and we have plenty of computing power for anything better than
bruteforce algorithms. But our psychology lags a lightyear behind technology:
we still hobble on mental crutches of irrelevant authority and peer support,
flawed analogies and needless experimentation.
Prize for contributions
I offer prizes up to a
total of $500K for debugging, optimizing, and extending this algorithm: github.
Contributions must fit into
incrementalcomplexity hierarchy outlined here. Unless you find a flaw in my
reasoning, which would be even more valuable. I can also pay monthly, but there
must be a track record.
Winners will have an option
to convert the awards into an interest in all commercial applications of a
final algorithm, at the rate of $10K per 1% share. This option is informal and
likely irrelevant, mine is not a commercial enterprise. Money can’t be primary
motivation here, but it saves time.
Winners so far:
2010: Todor Arnaudov, $600 for suggestion to buffer old inputs after
search. This occurred to me more than once before, but I rejected it as
redundant to potential elevation of these inputs. Now that he made me think
about it again, I realized that partial redundancy can preserve the detail at
much lower cost than elevation.
The buffer is accessed if
coordinates of contents get relatively close to projected inputs (that and
justification is mine). It didn’t feel right because brain has no substrate for
passive memory, but we do now.
2011: Todor, $400
consolation prize for understanding some ideas that were not clearly explained
here.
2016: Todor Arnaudov, $500
for multiple suggestions on implementing the algorithm, as well as for the
effort.
2017:
Alexander Loschilov, $2800 for help in converting my level 1 pseudo code into Python,
consulting on PyCharm and SciPy, and for insistence on 2D clustering,
FebruaryApril.
Todor Arnaudov: $2000
for help in optimizing level_1_2D, JuneJuly.
Kapil Kashyap: $ 2000 for stimulation and effort, help with
Python and level_1_2D, SeptemberOctober
2018:
Todor Arnaudov,
$1000 mostly for effort and stimulation, JanuaryFebruary
Andrei Demchenko, $1800 for conventional
refactoring in line_POC_introductory.py, interface improvement and few improvements
in the code, April  May.
Todor Arnaudov, $2000 for help in debugging frame_dblobs.py, September
 October.
Khanh Nguyen, $2700, is a lead programmer of this project starting in
December.
2019:
Stephan Verbeeck, $1000 for getting me to return to using gradient (vs.
orthogonal differences), and for his perspective on colors and line tracing,
January.
Khanh Nguyen, $3000, is a lead programmer of this project, JanuaryFebruary