Theory without practice is empty, practice without theory is blind

of two images of the same scene?

squares error

Correlation

find shifts
Success!

pre-aligned images
Match them by correlation
Construct local displacement field

image

Digital map

Correlation?

maximum

Mutual information: Ideal maximum

Unfortunately, it’s difficult to compute and not applicable to vector maps

Viola P.A. Alignment by Maximization of Mutual Information: PhD thesis, MIT, Cambridge, Massachusetts. 1995. 156 p.

group of transformations if it is not known?
How to construct contours and structural elements optimally?
How to choose the most adequate number of contours and structural elements?
Are precision criteria such as mean square error suitable? Or have they the same shortcomings as correlation?

not the most precise description!

extended for the task of matching structural elements
In general, the minimum description length can be used for model selection

The best model is the model that minimizes the sum
the description length (in bits) of the model,
the description length (in bits) of data encrypted with help of the model (deviation of data from model).

probability: P(H)
Likelihood: P(D | H)

The description length of the model: –log P(H)
The description length of data encrypted with the help of the model: –log P(D | H)

model is chosen as trade-off between precision and complexity

on dispersion

with the use of the minimum description length approach // Proc. SPIE. 2004. Vol. 5426. P. 164–175.

dynamic scenes

Essential for correct estimation of a dynamic scene structure

Wrong

Correct

A.N. Averkin, I.P. Gurov, M.V. Peterson, A.S. Potapov. Spectral-Differential Feature Matching and Clustering for Multi-body Motion Estimation // Proc. MVA2011 IAPR Conference on Machine Vision Applications. 2011. June 13-15, Nara, Japan. P. 173–176.

case of MDL)
…
Image analysis
Segmentation;
Object recognition and image matching;
Optical flow estimation;
Structural description of images;
Changes detection;
…
Learning in symbolic domains, etc.

Various applications of MDL

are calculated within heuristically defined coding schemes
Success of a method is highly determined by the utilized coding scheme
Is there some theory that overcomes this arbitrariness?

machine
K – Kolmogorov complexity,
l(H) – length of program H
H * – best description/model of data D

Two-part coding:

UTM defines the universal model space

OR

if H is probabilistic program

if full model is separated into two parts

probabilities dominates (with multiplicative factor) over any other distribution
Bayesian prediction with the use of these priors converges in limit with prediction based on usage of true distribution

Solomonoff, R.: Algorithmic Probability, Heuristic Programming and AGI. In: Baum, E., Hutter, M., Kitzelmann, E. (eds). Advances in Intelligent Systems Research, vol. 10 (proc. 3rd Conf. on Artificial General Intelligence), pp. 151–157 (2010).

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int a=10000,b,c=8400,d,e,f[8401],g;
main() {for(;b-c;)f[b++]=a/5;
for(;d=0,g=c*2;c-=14, printf("%.4d",e+d/a),e=d%a)
for(b=c;d+=f[b]*a,f[b]=d%--g,d/=g--,--b;d*=b);}

By D.T. Winter

No. 2: all emeralds are greu
(that is green before 2050, and blue after this time)
Likelihood of observation data equals
How can we calculate prior probabilities of these two hypotheses?

Is it possible to ground prior probabilities?

Probability theory allows to deduce one probability from another. But what are the initial probabilities?
Universal priors work

Solomonoff R. Does Algorithmic Probability Solve the Problem of Induction? // Oxbridge Research, P.O.B. 391887, Cambridge, Mass. 02139. 1997.

the source of overlearning/ overfitting/ oversegmentation, etc.
Why is any new narrow learning method “yet another classifier”
Why are feed forwards neural networks not really “universal approximators”
And at the same time, why is “no free lunch theorem” not true

arbitrary computable environment
incomputable or computationally infeasible
approximations are either inefficient or not universal
Practical methods
work in non-toy environments
set of environments is highly restricted
=> Bridging this gap is necessary

of the algorithmic probabilities on the choice of UTM appears to be very useful in order to put any prior information and to reduce necessary amount of training data
UTM contains prior information
=> UTM can be optimized to account for posterior information

MDL criterion

Essentially incorrect segments

identically distributed samples of random variable (no segmentation).
Image is divided into regions; brightness values described independently within each region.
Second order functions are fit in each region, and brightness residuals are described as iid random variables.
Mixes of Gabor functions are used as regression models.

paradigm of learnable computer vision algorithms based on the representational MDL principle // Proc. SPIE. 2010. V. 7696. P. 769606.

(contours and regions)

Structural descriptions (line segments, arcs, ellipses, corners, blobs)

Features

Keypoints

Composite structural elements

Knowledge-based

*Marr, D.: Vision: A Computational Investigation into the Human Representation and Processing of Visual Information. MIT Press (1982).

representations

Correlation-based methods

Maximization of mutual information

Invariant moments

Distance transform

Pattern recognition

Tree search

Formal grammars

Hough transform

Graph-theoretic methods

Decision trees

Rule bases

Logic inference

…

are calculated within heuristically defined coding schemes
Success of a method is highly determined by the utilized coding scheme
In computer vision and machine learning, some representation is used in every method
But how to construct the best representation?
Representations correspond to ‘coding schemes’ in MDL applications. They should also be constructed on the base of strict criterion
But from what space and how?

bit
Np=16

%(learn)=0.0
%(test)=18.4
L = 62.0 bit
Np=25

No outliers

Worst generalization!

on the covariance matrix of normal distribution

L=834
L=855
L=855

L=838
L=817
L=826

L=857
L=826
L=823

1 2 3

2 3 4

2 3 4

be applied to mass problems since typically

where K is Kolmogorov complexity on universal machine U

However, can hold

One can search for models

within some best representation

for each xi independently

Potapov, A., Rodionov, S.: Extending Universal Intelligence Models with Formal Notion of Representation. In: J. Bach, B. Goertzel, M. Iklé (Eds.): AGI’12, LNAI 7716, pp. 242–251 (2012).

be such the program S for UTM U that for any data entity D the description H exists that U(SH)=D.

Representational MDL principle

The best image description has minimum length within given representation
The best image representation minimizes summed description length of images from the given training set (and the length of representation itself).
Main advantage: applicable to any type of representation; representation is included into general criterion as a parameter.

For example, image analysis tasks are mass problems: the same algorithm is applied to different images (or patterns) independently.

Potapov A.S. Principle of Representational Minimum Description Length in Image Analysis and Pattern Recognition // Pattern Recognition and Image Analysis. 2012. V. 22. No. 1. P. 82–91.

recognition methods
Selecting a representation that better fits the training sample from a specific domain either from a family of representations or from a fixed set of hand-crafted representations
Improve data analysis methods for specific representations

* Potapov A.S. Synthetic pattern recognition methods based on the representational minimum description length principle // Proc. OSAV'2008. 2008. P. 354–362.

Peterson M. A Representational MDL Framework for Improving Learning Power of Neural Network Formalisms // IFIP AICT 381. Springer, 2012. P. 68–77.

Precision of forecasting depends on type of nonlinearity
ANN with 4 neurons, 11 connections, and 2 second-order connections: MSE=220 (typical MSE: 214–625*)

short-term prediction precision and the RMDL criterion in average, one can agree with the statement: “MSE and NMSE are not very good measures of how well the model captures the dynamics”

Test: Financial time series

description within more complex representation

Gurov I., Potapov A. Investigation of OCT Images Descriptions on the Base of Representational MDL Principle // Proc. MVA2009 IAPR Conference on Machine Vision Applications. P. 320–323.

207864

Description length, bits
S-0: 235566
S-1: 219641
S-2: 215066

Description length, bits
S-0: 236421
S-1: 213015
S-2: 206204

S-1: oversegmentation
S-2: correct detection of layers

S-1 and S-2 are almost the same (and plausible) detection of thin layers

Differing segmentation results for a single thick layer (light absorption with depth causes regular reduction of brightness). Some inclusions are not detected.

S-1: odd layer is detected and inclusion is missed
S-2: plausible results of segmentation

points using predefined hand-crafted feature transform

Example of some found linear feature transforms

Example of some feature transforms for another environment

feature transform

~50% of failures with predefined features were matched successfully with learned features (new images of the same environment were used)

elements

description

Potapov A.S. Theoretico-informational approach to the introduction of feedback into multilevel machine-vision systems // Journal of Optical Technology. 2007. Vol. 74. Iss. 10. P. 694–699.

the first image

Initial structural elements of the second image

Fixed structural descriptions: same for both images

These descriptions slightly less precise, but w.r.t. images, but only one of them can be used instead of two

efficient search and restricted families of representations
Deep learning
Not universal
Compact (one-level ANNs should be exponentially larger than multi-level ANNs to represent some concepts => particular case of RMDL)
Higher expressive power or more efficient search than those of former methods

representations

Efficient search

Universal search

Theory

Practice

problems
Generality should be achieved not by a single uniform method solving any problem in the same fashion, but by automatic construction of (non-universal) efficient methods
Program specialization is the appropriate concept*, which relates general and narrow intelligence methods
However, no analysis of possible specialization of concrete models of universal intelligence has been given yet.

* Khudobakhshov, V.: Metacomputations and Program-based Knowledge Representation. In:K.-U. Kühnberger, S. Rudolph, P. Wang (Eds.): AGI’13, LNAI 7999, pp. 70–77 (2013).

that can be much more efficient than p(x0, .)

Let pL(x,y) be some program (in some language L) with two arguments
Specializer specR is such program (in some language R) accepting pL and x0 that

Futamura-Turchin projections

same S
This search cannot be non-exhaustive for any S, but it can be efficient for some of them
One can consider computationally efficient projection spec(MSearch, S):

Universal mass induction consists of two procedures

Search for models

Search for representations

Potapov A., Rodionov S. Making Universal Induction Efficient by Specialization // B. Goertzel et al. (Eds.): AGI 2014. Lecture Notes in Artificial Intelligence. 2014. V. 8598. P. 133–142.

general techniques for exponential speedup exists
And how to get S? RSearch is still needed
Find S'=spec(MSearch(S, x), S*) simultaneously with S*
Main properties of S, S':

S is a generative representation (decoding)
S' is a descriptive representation (encoding)
S' is also the result of specialization of the search for generative models, so in general it can include some sort of optimized search
Simultaneous search for S and S' will be referred to as SS'-search

us to utilization of the MDL principle that was heuristically applied for solving many tasks
The MDL principle is very useful tool for introducing model selection criteria free from overfitting in the tasks of image analysis and pattern recognition
We introduced the representational MDL principle to bridge the gap between universal induction and practical methods and used it to extend practical methods
The remaining difference between universal and practical methods is in search algorithms. Specialization of universal search is necessary to automatically produce efficient methods