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Feature Selection as Relevant Information Encoding . Naftali Tishby School of Computer Science and Engineering The Hebrew University, Jerusalem, Israel NIPS 2001. Many thanks to: Noam Slonim Amir Globerson Bill Bialek Fernando Pereira Nir Friedman. Feature Selection?.

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Feature selection as relevant information encoding

Feature Selectionas Relevant Information Encoding

Naftali Tishby

School of Computer Science and Engineering

The Hebrew University, Jerusalem, Israel

NIPS 2001


Many thanks to noam slonim amir globerson bill bialek fernando pereira nir friedman

Many thanks to:

Noam Slonim

Amir Globerson

Bill Bialek

Fernando Pereira

Nir Friedman


Feature selection
Feature Selection?

  • NOT generative modeling!

    • no assumptions about the source of the data

  • Extracting relevant structure from data

    • functions of the data (statistics) that preserve information

  • Information about what?

  • Approximate Sufficient Statistics

  • Need a principle that is both general and precise.

    • Good Principles survive longer!


A simple example
ASimple Example...



A new compact representation
A new compact representation

The document clusters preserve the relevant

information between the documents and words


Documents

Words


Mutual information
Mutual information

How much X is telling about Y?

I(X;Y): function of the joint probability distribution p(x,y) -

minimal number of yes/no questions (bits) needed to ask about x, in order to learn all we can about Y.

Uncertainty removed about X when we know Y:

I(X;Y) = H(X) - H( X|Y) = H(Y) - H(Y|X)

I(X;Y)

H(X|Y)

H(Y|X)


Relevant coding
Relevant Coding

  • What are thequestionsthat we need to askaboutXin orderto learn about Y?

  • Need to partition X into relevant domains, or clusters, between which we really need to distinguish...

P(x|y1)

X|y1

y1

y2

X|y2

P(x|y2)

X

Y


Bottlenecks and Neural Nets

  • Auto association: forcing compact representations

  • is a relevant code of w.r.t.

Input

Output

Sample 1

Sample 2

Past

Future


  • Q: How many bits are needed to determine the relevant representation?

    • need to index the max number of non-overlapping green blobs inside the blue blob:

      (mutual information!)


  • The idea: find a compressed signal

    that needs short encoding ( small )

    while preserving as much as possible the information on the relevant signal ( )


A variational principle
A Variational Principle

We want a short representation of X that keeps the information about another variable, Y, if possible.


The self consistent equations
TheSelf Consistent Equations

  • Marginal:

  • Markov condition:

  • Bayes’ rule:


The emerged effective distortion measure:

  • Regular if is absolutely continuous w.r.t.

  • Small if predicts y as well as x:



The information bottleneck algorithm
The Information BottleneckAlgorithm

“free energy”


  • The Information - plane, the optimal for agiven is a concave function:

impossible

Possible phase


Manifold of relevance
Manifold of relevance

The self consistent equations:

Assuming acontinuous manifoldfor

Coupled (local in ) eigenfunction equations, with  as an eigenvalue.



Multivariate information bottleneck
Multivariate Information Bottleneck

  • Complex relationship between many variables

  • Multiple unrelated dimensionality reduction schemes

  • Trade between known and desired dependencies

  • Express IB in the language of Graphical Models

  • Multivariate extension of Rate-Distortion Theory


Multivariate Information Bottleneck:

Extending the dependency graphs

(Multi-information)


Sufficient dimensionality reduction with amir globerson
Sufficient Dimensionality Reduction(with Amir Globerson)

  • Exponential families have sufficient statistics

  • Given a joint distribution , find an approximation of the exponential form:

This can be done by alternating maximization of Entropy under the constraints:

The resulting functions are our relevant features at rank d.


Summary
Summary

  • We present a general information theoretic approach for extracting relevant information.

  • It is a natural generalization of Rate-Distortion theory with similar convergence and optimality proofs.

  • Unifies learning, feature extraction, filtering, and prediction...

  • Applications (so far) include:

    • Word sense disambiguation

    • Document classification and categorization

    • Spectral analysis

    • Neural codes

    • Bioinformatics,…

    • Data clustering based on multi-distance distributions


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