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 {E} \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 nonlinearity 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) = u e^{-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, $\mathbf {1}$ 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}^{C \times N}

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
         $\mathbf{w_p} \leftarrow \frac{1}{M}\mathbf{X} g(\mathbf{w_p}^T \mathbf{X})^T - \frac{1}{M}g'(\mathbf{w_p}^T\mathbf{X})\mathbf{1} \mathbf{w_p}$ 
         $\mathbf {w_{p}} \leftarrow \mathbf {w_{p}} -\sum _{j=1}^{p-1}\mathbf {w_{p}} ^{T}\mathbf {w_{j}} \mathbf {w_{j}}$ 
         $\mathbf{w_p} \leftarrow \frac{\mathbf{w_p}}{\|\mathbf{w_p}\|}$

 Output:  $\mathbf {W} ={\begin{bmatrix}\mathbf {w_{1}} ,\dots ,\mathbf {w_{C}} \end{bmatrix}}$ 

 
 Output:  $\mathbf {S} =\mathbf {W^{T}} \mathbf {X}$

References

1 2 Hyvärinen, A.; Oja, E. (2000). "Independent component analysis: Algorithms and applications" (PDF). Neural Networks. 13 (4–5): 411–430. doi:10.1016/S0893-6080(00)00026-5. PMID 10946390.
↑ Hyvarinen, A. (1999). "Fast and robust fixed-point algorithms for independent component analysis" (PDF). IEEE Transactions on Neural Networks. 10 (3): 626–634. doi:10.1109/72.761722. PMID 18252563.

External links

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