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Nonparametric information estimation using Rank based Euclidean Graph Optimization

methods

Barnabás Póczos

Department of Computing Science,

University of Alberta, Canada

Machine Learning Lunch

Carnegie Mellon University

Feb 8, 2010

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Outline

GOAL: Information estimation

Background on information and entropy

Applications in machine learning

Graph optimization methods

TSP, MST, kNNCopula transformation

Information estimation

Background on information and entropy

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Who cares about dependence?

• Observing physical, biological, chemical systems

– Which observations are dependent / independent?

– Is there dependence between inputs and outputs?

• Neuroscience: dependence between stimuli and neural response

• Stock markets– which stocks are dependent of each others, which are

independent,

– dependence on other events?

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What is dependence?

We want to measure dependence

Correlation?

… nope that’s not good enough…

Correlation can be zero when there is dependence between the variables

Isn’t this only an artificial problem?

It is a serious problem…

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Correlation?… nope that’s not good enough

Correlation measures only pairwise dependencies.

What can we do when we have more variables?

X1

X2

covariance matrix = I

cross-correlation = 0,

but there is dependence

that’s why PCA is not able to separate mixed sources, and we need

ICA methods for considering higher order dependencies, too.

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What is dependence ?

Alfréd Rényi

1961, Hungary

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-divergence

• Rényi’s -information

• Shannon’s mutual information

Special cases:

Theorem:

1967, Hungary

Imre Csiszár

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Measuring Dependence and Uncertainty

Shannon mutual information

Measuring uncertainty (Shannon entropy)

1948, Princeton

Claude Shannon

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Rényi’s information and entropy

Rényi’s entropy

Rényi’s information

Applications of mutual information in machine learning

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Feature selection (Peng et. al, 2005)

Goal:

• find a set of features that has the largest dependence

with the target class

• find dependent and independent features

XT =XT Rn x d

labels

Microarray dataprocessing

d = number of genes

n = number of patients

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Feature selection (Peng et. al, 2005)

XT =XT Rn x d

labels

d = number of genes

n = number of patients

Max dependency

Max relevance

Min redundancy, Max relevance

Don’t want dependent features

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Independent Subspace Analysis

Observation

Hidden, independent sources

(subspaces)

Estimate A and S observing samples from X only

Goal:

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Independent Subspace Analysis

In case of perfect

separation WA is a

block permutation

matrix

Objective:

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Mutual Information Applications

• Information theory, Channel capacity

• Information geometry

• Clustering

• Optimal experiment design, active learning

• Prediction of protein structure

• Drug design

• Boosting

• fMRI data processing

• …

The estimation problem

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The estimation problem

GOAL:

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How can we estimate these?

How can we estimate them?

Graph optimization methods

TSP, MST, kNN

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History• 1959, Beardwood, Halton, Hammersley

– Given uniform iid samples on [0,1]2. TSP length=?

• 1976, Karp

– randomly distributed cities

– This asymptotic can be used for developing a deterministic poly time near-optimal(1+) TSP algorithm (a.s.)

– 1st to show that 9 alg. for a stochastic version

of an NP complete problem (TSP) that inpolynomial time gives a near optimal solution (a.s.)

John Hammersley

Oxford, UK

Richard Karp

Berkeley, CA

– uniform iid samples on [0,1]d. TSP length=?

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• 1981 – TSP ) MST, Minimal Matching graphs, conjecture for kNN

• Renyi’s -entropy estimation?

observations with density f on [0,1]d

• Beardwood, Halton, Hammersley

Michael Steele Joseph Yukich Wansoo Rhee

1999, Hero, Costa & Michel

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Steele, Yukich theorem for MST, TSP, Minimal Matching

(Fig taken from J. Costa and A. Hero)Sensitive to outliers…!

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A new result: This procedure works on Set-Nearest Neighborhood

graphs, too

Calculate:

S={1,3,7} ) consider 1st, 3rd, 7th nearest neighbors of each node

S={k} ) consider the kth nearest neighbors of each node

S={1,2,…,k} ) kNN graphs

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Rényi’s entropy estimators using Set-Nearest Neighborhood graphs

(Fig. taken from J. Costa and A. Hero)

How can we get information estimators from entropy estimators?

How can we make the estimators more robust?

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The invariance trick

Information is preserved under monotonic transformations.

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Transformation to get uniform marginals

The transformation (copula transformation):

Monotone transform Uniform marginals

• Information estimation problem is reduced for

Rényi’s entropy estimation

• a little problem: we don’t know Fi distribution functions

Monotone transformation leading to uniform marginals?

Prob theory 101:

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Copula methods

• Very popular in financial analysis

– Capture dependence between marginal variables

– An alternative form of normalization

• … but not so widespread in machine learning…

Well in financial analysis, they led

to the global financial crisis…

Maybe we should use them in ML more often!

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The copula transformation

Monotone transform Uniform marginals

a little problem: we don’t know Fi distribution functions

Solution:

Use empirical distributions Fjn and empirical copula transform.

We need this in 1D only! ) no curse of dimensionality.

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Empirical copula transformation

We don’t know F1,…,Fd distribution functions

) estimate them with empirical distribution functions

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Empirical copula transformation

“true” copula transformation:

empirical copula transformation:

empirical copula transformation of the observations:

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REGO AlgorithmRank-based Euclidean Graph Optimization

Theoretical Results

consistency

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Is consistency trivial?

Difficulties:

• The true copula transformation is not known… only the empirical

• Samples lie on a grid, not coming from the true copula distribution

True copula transform

Empirical copula transform

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• Empirical copula transformed points are not iid in n

) Steele & Yukich’s theorem cannot be applied

We need different proofs for the two cases:

• 0<p· 1: We have triangle inequality for ||.||p, but it is not Lipschitz

• 1<p<d/2: ||.||p is Lipschitz, but no triangle inequality

Difficulties:

Is consistency trivial?

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Consistency Results

Main theorem:

Note:• Consistent MI estimator using ranks only

• Ranks only ) Robust

• We don’t have theoretical results for other d and values…

Empirical Results

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Empirical results, consistency in 2D

n

kMST=0.8n

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Empirical results, consistency in 10D

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Image registration

n=4900=70 x 70

200 outliers

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Independent subspace analysis

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Independent subspace analysisin the presence of outliers

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Infinitesimal robustness (finite sample influence function)

The finite sample influence functions

The amount of change caused by adding one outlier, x

(with some modification)

It cannot be arbitrarily big!

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Conclusion

Marriage of seemingly unrelated mathematical areas produced a consistent and robust

Rényi’s information estimator

Graph optimization methods

TSP, MST, kNNCopula transformation

Information estimation

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Thanks for your attention!

“Hey Rege, Róka Rege, Hey REGŐ Rejtem…

I am calling my grandfather, I feel his soul here.

I am listening to his words, they are breaking the silence.

Impart your knowledge to your grandson please,

1000 years have already passed…,

Knowledge equals life, hey!”