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Saturday, December 13 Morning session

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NIPS 2003 Workshop onInformation Theory and Learning:The Bottleneck and Distortion ApproachOrganizers:Thomas Gedeon Naftali Tishbyhttp://www.cs.huji.ac.il/~tishby/NIPS-Workshop

Saturday, December 13

Morning session

- 7:30-8:20 N. TishbyIntroduction – A new look at Shannon’s Information Theory
- 8:20-9:00 T. GedeonMathematical structure of Information Distortion methods
- 9:00-9:30 D. Miller Deterministic annealing
- 9:30-10:00 N. SlonimMultivariate Information Bottleneck
- 10:00-10:30 B. MumeyOptimal Mutual Information Quantization is NP-complete
Afternoon session

- 16:00-16:20 G. Elidan The Information Bottleneck EM algorithm
- 16:20-16:40 A. GlobersonSufficient Dimensionality Reduction with Side Information
- 16:40-17:00 A. ParkerPhase Transitions in the Information Distortion
- 17:00-17:20 A. Dimitrov Information Distortion as a model of sensory processing
break

- 17:40-18:00 J. SinkkonenIB-type clustering for continuous finite data
- 18:00-18:20 G. ChechikGIB: Gaussian Information Bottleneck
- 18:20-18:40 Y. CrammerIB and data-representation – Bregman to the rescue
- 18:40-19:00 B. WestoverAchievable rates for Pattern Recognition

The fundamental dilemma: Simplicity/Complexity versus Accuracy

- Lessons from Statistical Physics for Machine Learning
- Is there a “right level” of description?

- Shannon’s source and channel coding – dual problems
- [Mutual] Information as theFundamental Quantity
- Information Bottleneck (IB) andEfficient Representations
- Finite Data Issues

- Words, documents and meaning…
- Understanding neural codes …
- Quantization of side-information
- Relevant linear dimension reduction: Gaussian IB
- Using “irrelevant” side information
- Multivariate-IB and Graphical Models
- Sufficient Dimensionality Reduction (SDR)

y=f(x)

Tradeoff between accuracy and simplicity

Y

Good models should enable Prediction of new data…

X

Y

Cluster models?

X

Physical systems with many degrees of freedom:

What is the right level of description?

- “Microscopic Variables” – dynamical variables (positions, momenta,…)
- “Variational Functions” – Thermodynamic potentials
(Energy, Entropy, Free-energy,…)

- Competition between order (energy) and disorder (entropy)…

Emergent “relevant” description

Order Parameters

(magnetization,…)

Statistical models of complex systems:

What is the right level of description?

- “Microscopic Variables” – probability distributions (over all observed and hidden variables)
- “Variational Functions” – Log-likelihood, Entropy,… ?(Multi-Information, “Free-energy”,… )
- Competition between accuracy and complexity

Emerged “relevant” description

Features

(clusters, sufficient statistics,…)

Limited data

Bounded

Computation

Accuracy

Survivability

Possible Models/representations

Complexity

Efficiency

When there is a (relevant) prediction or distortion measure

Accuracy small average distortion (good predictions)

Complexity short description (high compression)

A general tradeoff between distortion and compression:

Shannon’s Information Theory

decoder

encoder

receiver

source

channel

Sent

messages

Received

messages

symbols

Source

coding

Channel

coding

Channel

decoding

Source

decoding

0110101001110…

Compression

Error Correction

Decompression

Source Entropy

Channel Capacity

Rate vs Distortion

Capacity vs Efficiency

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

(mutual information!)

- M1: For two nodes, X and Y, the shared information,I[X;Y], is a continuous function of p(x,y).
- M2: when one of the variables defines the other uniquely (1-1 mapping from X to Y) and both p(x)=p(y)=1/K , then I[X;Y] is a monotonic function of K.
- M3: If X is partitioned into two subsets, X1 and X2, then the amount of shared information is additive in the following sense: I[X;Y]=I[(X1,X2);Y]+p(x1)I[X;Y|X1]+p(x2)I[X;Y|X2]
- M4: Shared information is symmetric: I[X;Y]=I[Y;X].
- M5:
Theorem:

Q:What is the simplest representation – fewer bits(/sec) (Rate) for a given expected distortion (accuracy)?

A: (RDT, Shannon 1960) solve:

Q: What is the maximum number of bits(/sec) that can be sent reliably (prediction rate) for a given efficiency of the channel (power, cost)?

A: (Capacity-power tradeoff, Shannon 1956) solve:

- The tradeoff between expected representation size (Rate) and the expected distortion is expressed by the rate-distortion function:
- By introducing a Lagrange multiplier, , we have a variational principle:
- with:

The Rate-Distortion function, R(D), is the optimal ratefor agiven distortionand is a convex function.

R(D)

Achievable region

-bits/accuracy

impossible

D

0

The Capacity – Efficiency Tradeoff

bits/erg

Achievable Region

C(E)

Capacity at E

E (Efficiency (power))

R(D)

C(E)

E

D

In the case of a Gaussian channel (w. power constraint) and square distortion, double matching means:

And No coding!

“There isa curious and provocative duality between the properties of a source with a distortion measure and those of a channel … if we consider channels in which there is a “cost” associated with the different input letters…

The solution to this problem amounts, mathematically, to maximizing a mutual information under linear inequality constraint… which leads to a capacity-cost function C(E) for the channel…

In a somewhat dual way, evaluating the rate-distortion function R(D) for source amounts, mathematically, to minimizing a mutual Information … again with a linear inequality constraint.

This duality can be pursued further and is related to the duality between past and future and the notions of control and knowledge. Thus we may have knowledge of the past but cannot control it; we may control the future but have no knowledge of it.”

C. E. Shannon (1959)

Bottlenecks and Neural Nets

- Auto association: forcing compact representations
- provide a “relevant code” of w.r.t.

Input

Output

Sample 1

Sample 2

Past

Future

Bottlenecks in Nature…

(w E Schniedman, Rob de Ruyter, W Bialek, N Brenner)

- The idea: find a compressed variable
that enables short encoding of ( small )

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

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

- 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 I-projection of a distribution q(x) on a convex set of distributions L:

“free energy”

The IB emergenteffective distortion measure:

The IB

algorithm

IB Coding Theorem(Wyner 1975, Bachrach, Navot, Tishby 2003):

The IB variational problem provides a tight convex lower bound on the expected size of the achievable representations of X, that maintain at least of mutual information on the variable Y. The bound can be achieved by: .

Formally equivalent to “Channel coding with Side-Information” (introduced in a very different context by Wyner, 75).

The same bound is true for the dual channel coding problem and the optimal representation constitutes a “perfectly matched source-channel” and requires “no further coding”.

IB Algorithm Convergence(Tishby 99, Friedman, Slonim,Tishby 01):

The GeneralizedArimoto-Blauht IB algorithm converges to a local minima of the IB functional, (including its generalized multivariate case).

- Shannon’s Information Theory suggests a compelling framework
for quantifying the “Complexity-Accuracy” dilemma, by unifying

source and channel coding as a Min Max of mutual information.

- The IB method turns this unified coding principle into algorithms
for extracting “informative” relevant structures from empirical

joint probability distributions P(X1,X2)…

- The Multivariate IB extends this framework to extract
“informative” structures from multivariate distributions,

P(X1,…,Xn), via Min Max I(X1,…,Xn),and Graphical models

- IB can be further extended for
Sufficient Dimensionality Reduction

a method for finding informative continuous

low dimensional representations via Min Max I(X1,X2)

Many thanks to:

Fernando Pereira

William Bialek

Nir Friedman

Noam Slonim

Gal Chechik

Amir Globerson

Ran Bachrach

Amir Navot

Ilya Nemenman