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This repo contains misc explorations.

ICA vs flashier

We want to compare fastICA’s rank-1 update with that of flashier. Matthew previously examined an example where rank-1 fastICA almost perfectly picks up one column of the true mixing matrix, but flashier does not. Our conjecture is that flashier’s failure arises from an overemphasis on the fit term. We plan to investigate this further and potentially improve flashier’s rank-1 update.

fastICA as a Bayesian ICA with Rademacher prior on the sources

rank-1 FastICA can be considered as an EM algorithm with a Rademacher prior on the sources, with a spectral shift step to accelerate convergence (Särelä and Valpola 2005). Therefore, it is also equivalent to rank-1 EBCD with Rademacher prior. Consider the model for \(Y\in \mathbb{R}^{n\times d}\)

\[ Y_{ij} \sim \mathcal{N}(a_i s_j, \sigma^2), \quad s_i \sim \text{Rad}(1/2),\quad \sum_{i}^n a_i^2=1. \]

fastICA estimates \(\mathbf{s}=\begin{bmatrix}s_1 & \dots s_d\end{bmatrix}^\top\) by first estimating the “inverse” of \(\mathbf{a}\), via the following steps:

  • Pick an initial \(\mathbf{w}\)

  • Iterate

    • \(\mathbf{s}=Y^\top \mathbf{w}\)

    • Denoising: \(\mathbf{s}^+=f(\mathbf{s})\)

    • \(\mathbf{w}^+=Y \mathbf{s}^+\)

    • \(\mathbf{w}=\mathbf{w}^+ / \|\mathbf{w}^+\|\).

The first two steps are the E-step that constructs the estimated posterior mean \(\mathbb{E}[\mathbf{s} \mid Y, \mathbf{w}]\), and the last two steps are the M-step that maximize \(\mathbb{E}_q[\log p(\mathbf{s}, Y \mid \mathbf{w})]\).

Some observations

  • Fitting a parallel rank-\(K_0\) EB ICA model (\(K_0 > 1\)) is more likely to recover some of the true sources. However, a deflation scheme does not seem to work well, probably because of the initial poor rank-1 fit.

  • Whitening works well if we know the true rank.

Thoughts:

  • ICA/EBMF with different types of source?

  • Generalized EBMF? to different noise?

  • Is fastICA inability in unbalanced group due to centering or a wrong contrast function (logcosh is essentially a super Gaussian prior)?

  • fastICA seems to correspond to two different formulation of ICA: noiseless ICA with 1/cosh prior, and Bayesian ICA with Rademacher prior and Gaussian noise.

    • Relationship between 1/cosh and Rademacher?

  • fastICA treats \(E\) in \(LF^\top+E\) as Gaussian signals, but it doesn’t look for it in the first few factors.

Särelä, Jaakko, and Harri Valpola. 2005. “Denoising Source Separation.” Journal of Machine Learning Research 6 (9): 233–72. http://jmlr.org/papers/v6/sarela05a.html.