Independent Component Analysis

We imagine there is some data

s \in \mathbb{R}^d

that is generated via

d

independent sources.

We also have a mixing matrix

A \in \mathbb{R}^{d \times d}

that mixes the sources to produce the observed data

x \in \mathbb{R}^d

x = A s

Repeated observations of the data give us a set of training samples

\{x^{(1)}, \ldots, x^{(n)} \}

and we want to find the unmixing matrix

W = A^{-1}

that allows us to reconstruct the sources.

s = W x

We suppose that the distribution of each source

s_j

is given by

p_{s}(s_j)

and is independent of the other sources. Therefore the joint distribution of the sources is:

p(s) = \prod_{j=1}^d p_{s}(s_j)

Change of Variables in Probability

For any two vectors that are linearly related by

s = Wx

, the absolute value of the determinant of the transformation matrix

W

gives the factor by which the volume of any region in the space changes.

Therefore, if we have a probability distribution

p(s)

, the probability distribution of the transformed variables

p(x)

is given by:

p(s) = \frac{1}{|W|} \cdot p(x)

Since

s = W x

, the distribution of the sources in terms of the data is:

p(s)\cdot|W| = p(x)

Therefore the distribution of the data in terms of the sources is:

p(x) = \left( \prod_{j=1}^d p_{s}(w_j^T x) \right) \cdot |W|

We want to choose a monotonically increasing function that increases from

0

1

to be the CDF of our probability distribution. The derivative of the CDF will give us our probability density function. Choosing Sigmoid to be our CDF, we get:

p(s) = g'(s)

g(s) = \frac{1}{1 + \exp(-s)}

The log-likelihood of the data is then given by:

\ell(W) = \sum_{i=1}^n \left(\sum_{j=1}^d \log g'(w_j^T x^{(i)}) \right) + \log |W|

Taking the derivative of the log-likelihood with respect to

W

and setting it to 0, we get the following gradient descent update rule:

W \leftarrow W + \alpha \left( \left( \begin{array}{c} 1 - 2g(w_1^T x^{(i)}) \\ 1 - 2g(w_2^T x^{(i)}) \\ \vdots \\ 1 - 2g(w_d^T x^{(i)}) \end{array} \right) \left(x^{(i)}\right)^T + \left(W^{-1}\right)^T \right)

See derivation

Limitations

There are certain scenarios where Independent Component Analysis (ICA) might not work well:

If our unmixing matrix

W

is multiplied by a permutation matrix

P

, there is no way for us to know about it. In this case we won't be able to know which signal was from which source.

W \rightarrow PW

If a row in the unmixing matrix

W

is scaled by a constant

\alpha

, this will just result in the corresponding source being scaled by

1/\alpha

. There is no way for us to know if scaling has occurred. Therefore, we won't be able to retrieve the true amplitude of our signal.

If the data

x

follows a gaussian distribution, then our sources

s

will also follow a gaussian distribution. And gaussian distributions are symmetric in nature. Therefore, if our unmixing matrix

W

is multiplied by a rotation or reflection matrix

R

, there is no way for us to know about it.

W \rightarrow RW

Moreover, we have assumed that our data points are independent and identically distributed. This is however, not true for time-series data.

Despite all these limitations, ICA still works very well given enough data.

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