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Survey on ICA . Technical Report, Aapo Hyvärinen, 1999. http://ww.icsi.berkeley.edu/~jagota/NCS. Outline. 2nd-order methods PCA / factor analysis Higher order methods Projection pursuit / Blind deconvolution ICA definitions criteria for identifiability relations to other methods

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survey on ica

Survey on ICA

Technical Report, Aapo Hyvärinen, 1999.

http://ww.icsi.berkeley.edu/~jagota/NCS

outline

Outline

  • 2nd-order methods
    • PCA / factor analysis
  • Higher order methods
    • Projection pursuit / Blind deconvolution
  • ICA
    • definitions
    • criteria for identifiability
    • relations to other methods
    • Applications
  • Contrast functions
  • Algorithms
general model

General model

x = As + n

Observations

Mixing matrix

Noise

Latent variables, factors, independent components

find transformation

Find transformation

s = f (x)

Consider only linear transformation:

s = Wx

principal component analysis

Principal component analysis

  • Find direction(s) where variance of wTx is maximized.
  • Equivalent to finding the eigenvectors of C=E(xxT) corresponding to the k largest eigenvalues
factor analysis

Factor analysis

  • Closely related to PCA
  • x = As + n
  • Method of principal factors:
    • Assumes knowledge of covariance matrix of the noise: E(nnT)
    • PCA on: C = E(xxT)– E(nnT)
  • Factors are not defined uniquely, but only up to a rotation
higher order methods

Higher order methods

  • Projection pursuit
  • Redundancy reduction
  • Blind deconvolution
  • Requires assumption that data are not Gaussian
projection pursuit

Projection pursuit

  • Find direction w, such that wTx has an ’interesting’ distribution
  • Argued that interesting directions are those that show the least Gaussian distribution
differential entropy

Differential entropy

  • Maximised when f is a Gaussian density
  • Minimize H(wTx) to find projection pursuit directions (y = wTx)
  • Difficult to estimate the density of wTx
blind deconvolution

Blind deconvolution

Observe filtered version of s(t):

x(t) = s(t)*g(t)

Find filter h(t), such that

s(t) = h(t)*x(t)

example blind deconvolution

Example blind deconvolution

Seismic: ”statistical deconvolution”

ica definitions

ICA definitions

Definition 1 (General definition)

ICA of a random vector x consists of finding a linear transformation, s=Wx, so that the components, si, are as independent as possible, in the sense of maximizing some function F(s1,..,sm) that measure independence.

ica definitions1

ICA definitions

Definition 2 (Noisy ICA)

ICA of a random vector x consists of estimating the following model for the data:

x = As + n

where the latent variables si are assumed independent

Definition 3 (Noise-free ICA) x = As

statistical independence

Statistical independence

  • ICA requires statistical independence
  • Distinguish between statistically independent and uncorrelated variables
  • Statistically independent:
  • Uncorrelated:
identifiability of ica model

Identifiability of ICA model

  • All the independent components, but one, must be non-Gaussian
  • The number of observed mixtures must be at least as large the number of independent components, m >= n
  • The matrix A must be of full column rank
  • Note: with m < n, A may still be indentifiable
relations to other methods

Relations to other methods

  • Redundancy reduction
  • Noise free case
    • Find ’interesting’ projections
    • Special case of projection pursuit
  • Blind deconvolution
  • Factor analysis for non-Gaussian data
  • Related to non-linear PCA
applications of ica

Applications of ICA

  • Blind source separation
    • Cocktail party problem
  • Feature extraction
  • Blind deconvolution
objective contrast functions

Objective (contrast) functions

ICA method = Objective function + Optimization algorithm

  • Multi-unit contrast functions
    • Find all independent components
  • One-unit contrast functions
    • Find one independent component (at a time)
mutual information

Mutual information

  • Mutual information is zero if the yi are independent
  • Difficult to estimate, approximations exist
mutual information 2

Mutual information (2)

  • Alternative definition
mutual information 3

Mutual information (3)

H(X|Y)

H(Y|X)

I(X,Y)

H(X)

H(Y)

non linear pca

Non-linear PCA

  • Add non-linearity function g(.) in the formula for PCA
one unit contrast functions

One-unit contrast functions

  • Find one vector, w, so that wTx equals one of the independent components, si
  • Related to projection pursuit
  • Prior knowledge of number of independent components not needed
negentropy

Negentropy

  • Difference between differential entropy of y and differential entropy of Gaussian variable with same variance
  • If the yi are uncorrelated, the mutual information can be expressed as
  • J(y) can be approximated by higher-order cumulants, but estimation is sensitive to outliers
algorithms

Algorithms

  • Have x=As, want to find s=Wx
  • Preprocessing
    • Centering of x
    • Sphering (whitening) of x
      • Find transformation; v=Qx such that E(vvT)=I
      • Found via PCA / SVD
  • Sphering does not solve problem alone
algorithms 2

Algorithms (2)

  • Jutten-Herault
    • Cancel non-linear cross-correlations
    • Non-diagonal terms of W are updated according to
    • The yi are updated iteratively as y = (I+W)-1x
  • Non-linear decorrelation
  • Non-linear PCA
  • FastICA, ..., etc.
summary

Summary

  • Definitions of ICA
  • Conditions for identifiability of model
  • Relations to other methods
  • Contrast functions
    • One-unit / multi-unit
    • Mutual information / Negentropy
  • Applications of ICA
  • Algorithms
future research

Future research

  • Noisy ICA
  • Tailor-made methods for certain applications
  • Use of time correlations if x is a stochastic process
  • Time delays/echoes in cocktail-party problem
  • Non-linear ICA