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Independent Component Analysis

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Independent Component Analysis

An Introduction

Zhen Wei, Li Jin, Yuxue Jin

Department of Statistics

Stanford University

- Introduction
- History, Motivation and Problem Formulation

- Algorithms
- Stochastic Gradient Algorithm
- FastICA
- Ordering Algorithm

- Applications
- Concluding Remark

- There has been a wide discussion about the application of Independence Component Analysis (ICA) in Signal Processing, Neural Computation and Finance, first introduced as a novel tool to separate blind sources in a mixed signal. The Basic idea of ICA is to reconstruct from observation sequences the hypothesized independent original sequences.

- Similarity
- Feature extraction
- Dimension reduction

- Difference
- PCA uses up to second order moment of the data to produce uncorrelated components
- ICA strives to generate components as independent as possible

- Suppose that there are k unknown independent sources
- A data vector x(t) is observed at each time point t, such that
where A is a full rank scalar matrix

Independent

components

Observed

sequences

Recovered independent components

…

…

Mixing

process

A

Blind

Source

De-mixing

process

W

- The goal of ICA is to find a linear mapping W such that the unmixed sequences u
are maximally statistically independent

- Find some
where C is a diagonal matrix and P is a permutation matrix.

- The fundamental restriction in ICA is that the independent components must be nongaussian for ICA to be possible.
- This is because gaussianity is invariant under orthogonal transformation and hence make the matrix A not identifiable for gaussian independent components.

- Kurtosis
- Kurtosis can be very sensitive to outliers, when its value has to be estimate from a measured sample.

- Negentropy
- A guassian variable has the largest entropy among all random variables of equal variance.
- Definition:
where is entropy

and ygauss is a gaussian random variable of the same covariance matrix as y

- Mutual information
- Definition:
- Mutual information is a natural measure of the dependence between random variables.
- It is always non-negative, and zero if and only if the variables are statistically independent.

- If we constrain yi to be uncorrelated and of unit variance
where C is a constant that does not depend on W.

- This shows that finding an invertible transformation W that minimizes the mutual information is equivalent to finding directions in which the negentropy is maximized.

- Maximum likelihood Bell and Sejnowski (1995)
- Maximum entropy
- Minimum mutual information

- Low-Complexity Coding and Decoding (LOCOCODE ) Sepp Hochreiter et al. (1998)
- Neuro-mimetic approach

- The log-likelihood is:
where the fi are the density functions of the si

- Connection to mutual information:
if the fi were equal to the true distributions of

- Initialize the weight matrix W
- Iteration:
where is the learning rate, g is a nonlinear function, e.g.

- Repeat until converges to

- The ICAs are the components of

- Centering:
- Make the x-s mean 0 variables

- Whitening
- Transform the observed vector x linearly so that it has unit variance:
- One can show that:
where

- Initialize the weight matrix W
- Iteration:
where

- Repeat until convergence

- The ICAs are the components of

- Unlike PCA which has well-defined and intuitive explanation of the ordering of its components, i.e. the eigen values of its covariance matrix, ICA, however, deserves further investigation on this particular problem since a particular kind of ordering is not readily at hand.
- Follow a heuristic scheme called: testing-and-acceptance (TNA)

- Feature extraction: Recognize the pattern of excess returns of Mutual Funds in the financial market of China
- Data: the time series of excess returns of four mutual funds in the financial market of China

- Image de-noising
- ICA
- Sparse Code Shrinkage

- The example is exacted from (Hyvarinen, 1999).

- Suppose a noisy image model holds:
where n is uncorrelated noise.

where W is an orthogonal matrix that is the best orthogonal approximation of the inverse of the ICA mixing matrix.

- Sparse code shrinkage transformation:
Function g(.) is zero close to the origin and linear after a cutting value depending on the parameters of the Laplacian density and the Gaussian noise density.

1. Original image

2. Corrupted with noise

3. Recover by ICA and Sparse Code Shrinkage

3. Recover by classical wiener filtering

- ICA is a very flexible and widely-applicable tool which searches the linear transformation of the observed data into statistically maximally independent components
- It is also interesting to note that the methods to compute ICA: maximum negentropy, minimum Mutual Information, maximum likelihood are equivalent to each other (at least in the statistical sense). There is also resemblance between the forms of the gradient descent (Newton Raphson) algorithm and the FastICA algorithm.
- Other application prospects: audio (signal) processing, image processing, telecommunication, Finance, Education

[1] Amari, S., Cichocki, A., and Yang, H. (1996). A New Learning Algorithm for Blind Signal Separation, Advances in Neural Information Processing Systems 8, pages 757-763.

[2] Bell, A. J. and Sejnowski, T. J. (1995). An Information-Maximization Approach to Blind Separation and Blind Deconvolution. Neural Computation, 7:1129-1159

[3] Cardoso, J. and Soloumiac, A. (1993). Blind beamforming for non-Gaussian signals. IEEE Proceedings-F, 140(46):362-370.

[4] Chatfield, C. (1989). Analysis of Time Series: An Introduction, Fourth Edition. London: Chapman and Hall.

[5] Moulines, E., Cardoso, J.-F., and Cassiat, E. (1997). Maximum likelihood for blind separation and deconvolution of noisy signals using mixture models. Proc. ICASSP’ 97, volume 5, pages 3617-3620, Munich.

[6] Nadal, J.-P. and Parga, N. (1997). Redundancy reduction and independent component analysis: Conditions on cumulants and adaptive approaches. Neural Computation, 9:1421-1456.

[7] Xu, L., Cheung, C., Yang, H., and Amari, S. (1997). Maximum equalization by entropy maximization and mixture of cumulative distribution functions. Proc. Of ICNN’97, pages 1821-1826, Houston

[8] Yang, H., Amari, S., and Cichocki, A. (1997). Information back-propagation for blind separation of sources from non-linear mixtures. Proc. of ICNN, pages 2141-2146, Houston