1. Independent Component Analysis
CMPUT 466/551
Nilanjan Ray

2. The Origin of ICA: Factor Analysis
Multivariate data are often thought to be indirect measurements arising from some underlying sources, which cannot be directly measured/observed.
Examples
Educational and psychological tests use the answers to questionnaires to measure the underlying intelligence and other mental abilities of subjects
EEG brain scans measure the neuronal activity in various parts of the brain indirectly via electromagnetic signals recorded at sensors placed at various positions on the head.
Factor analysis is a classical technique developed in statistical literature that aims at identifying these latent sources.
Independent component analysis (ICA) is a kind of factor analysis that can uniquely identify the latent variables.

3. Latent Variables and Factor Analysis
Latent variable model:
or,
Observed variable
Latent components
Mixing matrix
Factor analysis attempts to find out both the mixing coefficients and the
latent components given some instances of observed variables

4. Latent Variables and Factor Analysis…
Typically we require the latent variables to have unit variance and to be uncorrelated.
Thus, in the following model, cov(S) = I.
This representation has an ambiguity. Consider, for example an orthogonal matrix R:
So, is also a factor model with unit variance, uncorrelated latent variables.
Classical factor analysis cannot remove this ambiguity; ICA can remove this ambiguity.

5. Classical Factor Analysis
Model:
’s are zero mean, uncorrelated Gaussian noise.
q < p, i.e., the number of underlying latent factor is assumed less than
the number of observed components.
Diagonal matrix
The covariance matrix takes this form:
Maximum likelihood estimation is used to estimate A.
However, still the previous problem of ambiguity remains here too…

6. Independent Component Analysis
Step 1: Center data:
Step 2: Whiten data: compute SVD of the centered data matrix
After whitening in the factor model, the covariance of x, cov(x) = I, and A become orthogonal
Step 3: Find out orthogonal A and unit variance, non-Gaussian and independent S
PCA

7. Example: PCA and ICA Model:
Blind source separation (cocktail party problem)

8. PCA vs. ICA PCA: Find projections to minimize reconstruction error
Variance of projected data is as large as possible
2nd-order statistics needed (cov(x))
ICA:
Find “interesting” projections
Projected data look as non-Gaussian, independent as possible
Higher-order statistics needed to measure degree of independence

9. Computing ICA
Model:
Step 3: Find out orthogonal A and unit variance, non-Gaussian and independent S.
The computational approaches are mostly based on information theoretic criterion.
Kullback-Leibler (KL) divergence
Negentropy
Another different approach emerged recently is called “Product Density Approach”

10. ICA: KL Divergence Criterion
x is zero-mean and whitened
KL divergence measures “distance” between two probability densities
Find A such that KL(.) is minimized:
Joint density
Before whitening, ICA means finding components as non-Gaussian as possible
After whitening procedure: Cov(X) = I, ICA means finding components as independent as possible.
Independent density
H is differential entropy:

11. ICA: KL Divergence Criterion…
Theorem for random variable transformation says:
So,
Verify formula: why I got + log |det(A)| instead of –ve log |det(A)|
Hence,
Minimize with respect to orthogonal A

12. ICA: Negentropy Criterion
Differential entropy H(.) is not invariant to scaling of variable
Negentropy is a scale-normalized version of H(.):
Negentropy measures the departure of a r.v. s from a Gaussian r.v. with same variance
Optimization criterion:
I would think that normalized entropy would be better if (gaussian) noise is present. (not sure if the claim is true)

13. ICA: Negentropy Criterion…
Approximate the negentropy from data by:
FastICA ( is based on negentropy. Free software in Matlab, C++, Python…

14. ICA Filter Bank for Image Processing
An image patch is modeled as a weighted sum of basis images (basis functions):
Image patch
Basis functions (a.k.a. ICA filter bank)
Rows of AT are filters
Columns of A are filters
Filter responses
Jenssen and Eltoft, “ICA filter bank for segmentation of textured images,” 4th International symposium on ICA and BSS, Nara, Japan, 2003

15. Texture and ICA Filter Bank
Training textures
12x12 ICA basis functions or ICA filters
Jenssen and Eltoft, “ICA filter bank for segmentation of textured images,” 4th International symposium on ICA and BSS, Nara, Japan, 2003

16. Segmentation By ICA FB ICA Filter Bank With n filters I1, I2,…, In
These
are filter
responses
Image, I
Segmented image, C
Clustering
Above is an unsupervised setting.
Segmentation (i.e., classification in this context) can also be performed by
a supervised method on the output feature images I1, I2 , …, In.
A texture image
Segmentation
Jenssen and Eltoft, “ICA filter bank for segmentation of textured images,” 4th International symposium on ICA and BSS, Nara, Japan, 2003

17. On PCA and ICA PCA & ICA differ in choosing projection directions:
Different principle: least-square (PCA), independence (ICA)
For data compression, PCA would be a good choice
For discovering structures of data, ICA would be a reasonable choice