FastICA

FastICA is an efficient and popular algorithm for independent component analysis invented by Aapo Hyvärinen at Helsinki University of Technology.^[1]^[2] Like most ICA algorithms, FastICA seeks an orthogonal rotation of prewhitened data, through a fixed-point iteration scheme, that maximizes a measure of non-Gaussianity of the rotated components. Non-gaussianity serves as a proxy for statistical independence, which is a very strong condition and requires infinite data to verify. FastICA can also be alternatively derived as an approximative Newton iteration.

Algorithm

Prewhitening the data

Let the $\mathbf {X} :=(x_{ij})\in \mathbb {R} ^{N\times M}$ denote the input data matrix, $M$ the number of columns corresponding with the number of samples of mixed signals and $N$ the number of rows corresponding with the number of independent source signals. The input data matrix $\mathbf {X}$ must be prewhitened, or centered and whitened, before applying the FastICA algorithm to it.

Centering the data entails demeaning each component of the input data $\mathbf {X}$ , that is,

x_{ij}\leftarrow x_{ij}-{\frac {1}{M}}\sum _{j^{\prime }}x_{ij^{\prime }}

for each

i=1,\ldots ,N

and

j=1,\ldots ,M

. After centering, each row of

\mathbf {X}

has an expected value of

0

.

Whitening the data requires a linear transformation $\mathbf {L} :\mathbb {R} ^{N\times M}\to \mathbb {R} ^{N\times M}$ of the centered data so that the components of $\mathbf {L} (\mathbf {X} )$ are uncorrelated and have variance one. More precisely, if $\mathbf {X}$ is a centered data matrix, the covariance of $\mathbf {L} _{\mathbf {x} }:=\mathbf {L} (\mathbf {X} )$ is the $(N\times N)$ -dimensional identity matrix, that is,

\mathrm {E} \left\{\mathbf {L} _{\mathbf {x} }\mathbf {L} _{\mathbf {x} }^{T}\right\}=\mathbf {I} _{N}

A common method for whitening is by performing an eigenvalue decomposition on the covariance matrix of the centered data

\mathbf {X}

,

E\left\{\mathbf {X} \mathbf {X} ^{T}\right\}=\mathbf {E} \mathbf {D} \mathbf {E} ^{T}

, where

\mathbf {E}

is the matrix of eigenvectors and

\mathbf {D}

is the diagonal matrix of eigenvalues. The whitened data matrix is defined thus by

\mathbf {X} \leftarrow \mathbf {D} ^{-1/2}\mathbf {E} ^{T}\mathbf {X} .

Single component extraction

The iterative algorithm finds the direction for the weight vector $\mathbf {w} \in \mathbb {R} ^{N}$ that maximizes a measure of non-Gaussianity of the projection $\mathbf {w} ^{T}\mathbf {X}$ , with $\mathbf {X} \in \mathbb {R} ^{N\times M}$ denoting a prewhitened data matrix as described above. Note that $\mathbf {w}$ is a column vector. To measure non-Gaussianity, FastICA relies on a nonquadratic nonlinear function $f(u)$ , its first derivative $g(u)$ , and its second derivative $g^{\prime }(u)$ . Hyvärinen states that the functions

f(u)=\log \cosh(u),\quad g(u)=\tanh(u),\quad {\text{and}}\quad {g}'(u)=1-\tanh ^{2}(u),

are useful for general purposes, while

f(u)=-e^{-u^{2}/2},\quad g(u)=ue^{-u^{2}/2},\quad {\text{and}}\quad {g}'(u)=(1-u^{2})e^{-u^{2}/2}

may be highly robust.^[1] The steps for extracting the weight vector $\mathbf {w}$ for single component in FastICA are the following:

Randomize the initial weight vector $\mathbf {w}$
Let $\mathbf {w} ^{+}\leftarrow E\left\{\mathbf {X} g(\mathbf {w} ^{T}\mathbf {X} )^{T}\right\}-E\left\{g'(\mathbf {w} ^{T}\mathbf {X} )\right\}\mathbf {w}$ , where $E\left\{...\right\}$ means averaging over all column-vectors of matrix $\mathbf {X}$
Let $\mathbf {w} \leftarrow \mathbf {w} ^{+}/\|\mathbf {w} ^{+}\|$
If not converged, go back to 2

Multiple component extraction

The single unit iterative algorithm estimates only one weight vector which extracts a single component. Estimating additional components that are mutually "independent" requires repeating the algorithm to obtain linearly independent projection vectors - note that the notion of independence here refers to maximizing non-Gaussianity in the estimated components. Hyvärinen provides several ways of extracting multiple components with the simplest being the following. Here, $\mathbf {1_{M}}$ is a column vector of 1's of dimension $M$ .

Algorithm FastICA

Input:

C

Number of desired components

Input:

\mathbf {X} \in \mathbb {R} ^{N\times M}

Prewhitened matrix, where each column represents an

N

-dimensional sample, where

C<=N

Output:

\mathbf {W} \in \mathbb {R} ^{N\times C}

Un-mixing matrix where each column projects

\mathbf {X}

onto independent component.

Output:

\mathbf {S} \in \mathbb {R} ^{C\times M}

Independent components matrix, with

M

columns representing a sample with

C

dimensions.

 for p in 1 to C:
     $\mathbf {w_{p}} \leftarrow$  Random vector of length N
    while  $\mathbf {w_{p}}$  changes
         $\mathbf {w_{p}} \leftarrow {\frac {1}{M}}\mathbf {X} g(\mathbf {w_{p}} ^{T}\mathbf {X} )^{T}-{\frac {1}{M}}g'(\mathbf {w_{p}} ^{T}\mathbf {X} )\mathbf {1_{M}} \mathbf {w_{p}}$ 
         $\mathbf {w_{p}} \leftarrow \mathbf {w_{p}} -\sum _{j=1}^{p-1}(\mathbf {w_{p}} ^{T}\mathbf {w_{j}} )\mathbf {w_{j}}$ 
         $\mathbf {w_{p}} \leftarrow {\frac {\mathbf {w_{p}} }{\|\mathbf {w_{p}} \|}}$ 

 output  $\mathbf {W} \leftarrow {\begin{bmatrix}\mathbf {w_{1}} ,\dots ,\mathbf {w_{C}} \end{bmatrix}}$ 

 output  $\mathbf {S} \leftarrow \mathbf {W^{T}} \mathbf {X}$