独立成分分析

Independent Components Analysis

目录

• Independent Components Analysis
  • Ambiguity
  • Densities and Linear Transformations
  • ICA

Ambiguity

ICA is ambiguous to scaling and permutation. but usually it doesn't matter.

As long as the data is not Gaussian, it's possible, given enough data, to recover the n independent sources.

Densities and Linear Transformations

\[p_x(x)=p_s(Wx)\,|W| \]

ICA

\[p(x)=\prod_{i=1}^n p_s(w_i^Tx)\,|W| \]

s is source and x is the detection. and \(x=As=W^{-1}s\)

we assume the CDF of \(s\) is \(g(s)\) is sigmoid. the log likelihood is given by

\[\ell(W)=\sum_{i=1}^m(\sum_{j=1}^n \log g'(w_j^Tx^{(i)})+\log|W|) \]

m training samples, and n independent sources of voice.

the stochastic gradient ascent process is:

\[W := W +\alpha([1-2g(w_1^T x^{(i)});...;1-2g(w_n^T x^{(i)})]x^{(i)^T}+(W^T)^{-1}) \]