ICA Fixed-Point Algorithm (FastICA)

Application Background

The algorithm is derived from fixed-point recursive methodology and applies to any data type. Its emergence enables ICA analysis of high-dimensional data. Known as the Fixed-Point algorithm, it was proposed by Hyvärinen and colleagues at University of Helsinki. FastICA is a rapid optimization iterative algorithm utilizing batch processing, where substantial sample data participates in each iteration. From distributed parallel processing perspective, it can still be classified as a neural network algorithm.

FastICA exists in multiple implementations including fourth-order cumulant-based, maximum likelihood-based, and maximum negentropy-based forms. The algorithm employs fixed-point iterative optimization, ensuring faster and more robust convergence. FastICA completes in three stages: First, mean removal from observed signals; Second, whitening processed applied to mean-removed signals. These preliminary steps simplify subsequent ICA processing. Finally, independent component extraction follows the detailed flowchart.

Key Technologies

1) Mean removal constitutes the most fundamental preprocessing step in ICA algorithms. This process subtracts the mean vector from observations, converting signals to zero-mean variables. This preprocessing simplifies ICA computations without preventing mean estimation.

2) Typically acquired data exhibits correlation, requiring preliminary whitening or sphering processing. Whitening removes inter-signal correlations, simplifying subsequent independent component extraction. Generally, whitened data demonstrates improved algorithm convergence and enhanced stability compared to unprocessed data.

3) Multiple independent component estimation requires extension of maximum non-Gaussianity methods. Vectors corresponding to different components should be orthogonal in whitened space. Algorithm step 6 employs symmetric orthogonalization to ensure distinct signal separation, though estimation errors from the first vector may propagate to subsequent vectors.

FastICA output vectors may exhibit order permutation and amplitude variation due to two inherent uncertainties:

1) Output vector ordering uncertainty: inability to determine which extracted signal corresponds to which original source component.

2) Output signal amplitude uncertainty: inability to recover original source amplitudes.

However, since primary information resides in output signals, these uncertainties don't hinder practical applications. For deeper comprehension, FastICA operation can be detailed as follows:

1) For an m-sensor observation matrix X, perform mean removal to obtain zero-mean matrix X0.

2) Apply whitening transformation: Convert X0 to Y using Y=WX0, where W represents the whitening matrix.

3) Perform independent component analysis on matrix Y to obtain matrix A, where A=W^-1.

Code Implementation Insight: The whitening matrix W is typically computed via eigenvalue decomposition of the covariance matrix. The fixed-point iteration for component extraction involves non-linear contrast function optimization through approximate Newton iterations.

Through these three stages, we obtain matrix A where each column represents an independent component. These components are mutually independent and contain complete original signal information. Consequently, FastICA demonstrates broad application prospects in signal processing domains.