Source Separation (ICA)

Chapters

General Terms and tools
PCA
- PCA
- Hebbian Learning
- Kernel-PCA
Source Separation
- ICA
- Infomax ICA
- Second Order Source Separation
- FastICA
Stochastic Optimization
Clustering
- k-means Clustering
- Pairwise Clustering
- Self-Organising Maps
- Locally Linear Embedding
Estimation Theory
- Density Estimation
  - Kernel Density Estimation
  - Parametric Density Estimation
- Mixture Models - Estimation Models

Independent Component Analysis (ICA)

ICA allows the reconstruction of mixed signals. This could for example be multiple speakers on one audio track.

Requirements

Needs some prior knowledge
The number of sources to be recovered is a parameter to the algorithm. It is possible to choose a higher number of sources and afterwards remove sources that are only noise.

Limitations

sources must have non-Gaussian distributions
source amplitudes cannot be recovered

$p$ number of observations

$N$ number of dimensions

$p * N + N^{2}$ unknowns

Syntax

$\underset{―}{x}$ observations

$\underset{―}{A}$ Mixing Matrix (must be invertible for ICA to work)

$\underset{―}{s}$ Sources

$\hat{\underset{―}{s}}$ Recovered/Estimated Sources

$\underset{―}{W}$ Unmixing matrix

General principal

\underset{―}{x} = \underset{―}{A} \cdot \underset{―}{s} ⟹ \hat{\underset{―}{s}} = \underset{―}{W} \cdot \underset{―}{x}

Different methods leverage different prior knowledge

Cost functions

Vanishing cross-correlation functions (QDIAG, FFDIAG) (more or less invented by the institute)
non-Gaussianity (fastICA)
Statistical Independence $D_{K L} (P_{\underset{―}{s}} (\hat{\underset{―}{s}}))$
Infomax $H (\hat{\underset{―}{u}})$

Infomax ICA

Variables

See above
$P$ Probability distribution of th target space
$H$ our cost function
${\hat{u}}_{i} = {\hat{f}}_{i} (\sum_{j} w_{i j} x_{j})$
$\hat{u}$ is the vector with the results of our approximation function
${\hat{f}}_{i}$ is a freely chosen sigmoid cdf function that depends on ${\hat{s}}_{i}$ . $| {\hat{f}}_{i}^{'} ({\hat{s}}_{i}) | = {\hat{P}}_{s_{i}} ({\hat{s}}_{i}) ⟹ {\hat{f}}_{i} ({\hat{s}}_{i}) = \int_{- \infty}^{{\hat{s}}_{i}} d y {\hat{P}}_{s_{i}} (y)$
$e^{(α)}$ training error of a specific datapoint.

Infomax ICA uses empirical risk minimization(ERM) to do gradient ascent.

Example for ${\hat{f}}_{i}$

\begin{aligned} {\hat{f}}_{(y)} & = \frac{1}{1 + e x p (- y)} \\ \frac{{\hat{f}}_{(y)}^{″}}{{\hat{f}}_{(y)}^{'}} & = 1 - 2 {\hat{f}}_{(y)} \end{aligned}

We want to maximize $E^{G}$ which relies on the real distribution of the data. Since this distribution is unkown we use the proxy error function $E^{T}$ . $E^{T}$ uses a sum over all the data points instead of an integral over the real distribution. Otherwise the two are identical.

\begin{aligned} E^{G} & = l n | d e t \underset{―}{W} | + \int d \underset{―}{x} P_{\underset{―}{x}} (\underset{―}{x}) {\sum_{l = 1}^{N} l n ({\hat{f^{'}}}_{l}) (\sum_{k = 1}^{N} w_{l k} x_{k})} \\ E^{T} & = l n | d e t \underset{―}{W} | + \frac{1}{p} \sum_{α = 1}^{p} {\sum_{l = 1}^{N} l n ({\hat{f^{'}}}_{l}) (\sum_{k = 1}^{N} w_{l k} x_{k})} \end{aligned}

The Infomax cost function Infomax is the measure of how much mutual information two random variables have. It can also be expressed as the KL divergence(is equivalent?). We try to minimize the mutual information, because we believe that our sources are independent This is why we need to maximize the cost function

H = - \int d \hat{\underset{―}{u}} P_{\underset{―}{u}} (\hat{\underset{―}{u}}) l n P_{\underset{―}{u}} \hat{\underset{―}{u}} \overset{!}{=} m a x

Possible exam task Independent source signals work better if they match ${\hat{f}}_{i}$ .

Find a suitable ${\hat{f}}_{i}$

Gradient ascent for w

Batch Learning $Δ w_{i j} = \frac{\partial E^{T}}{\partial w_{i j}} = \frac{ϵ}{p} \sum_{α}^{p} \frac{\partial e^{(α)}}{\partial w_{i j}}$
Online Learning $Δ w_{i j} = ε \frac{\partial e^{(α)}}{\partial w_{i j}}$ Taylor approximation used because inverse is hard to find.

Natural Gradient

The natural gradient is faster than normal gradient descent.

While normal gradient changes the parameters by a fixed rate, the natural gradient changes the outcome distribution by a constant distance. This distance in distributions is measured by the KL Divergence.¹

Step size is normalized at each step $d \underset{―}{Z} = d \underset{―}{W} \cdot {\underset{―}{W}}^{- 1}$

Second Order (Blind)Source Separation

SOBSS allows the separation of sources after temporal shift or some other form of noise. sources are not iid but assumed to be time correlated / sequential.

Principle (As ambiguous as necessary because I just transcribed this in the lecture) Try different $w$ so that $\hat{s}$ matches in height.

Parameters

$τ$ time shift

Cost function minimization depends on the algorithm used on top.

For additional robustness remove $τ = 0$ from the set of possible solutions. This avoids some trivial solutions.

FastICA

Prerequisites

Whitened data
PCA
Kurtosis is known (prior knowledge)

Limitations

sensitive to outliers

Additional Parameters

$\underset{―}{u}$

$\tilde{\underset{―}{A}}$ is orthogonal matrix.

Find the maximum negentropy based on contrast function.

\begin{aligned} | k u r t (\hat{s}) | & \overset{!}{=} m a x_{\underset{―}{z}} \\ s . t . \\ v a r (\hat{s}) = z_{1}^{2} + z_{2}^{2} & \overset{!}{=} 1 \\ k u r t (\hat{s}) & = z_{1}^{4} + z_{2}^{4} \end{aligned}

We can show that

{\underset{―}{z}}_{o p t} = (\begin{matrix} 0 \\ \pm 1 \end{matrix}) or {\underset{―}{z}}_{o p t} = (\begin{matrix} \pm 1 \\ 0 \end{matrix})

Optimize for curtosis

Batch learning
Online Learning

Batch Learning

loop
- Initialize $\underset{―}{b}$ with random vector of unit length
- $δ \underset{―}{b} = ε s g n [k u r t ({\underset{―}{b}}^{T} \underset{―}{u})] ⟨ \underset{―}{u} ({\underset{―}{b}}^{T} u)^{3} ⟩$
- normalize $\underset{―}{b}$

Kurtosis

<0: sub-gaussian. (looks like rectangle) uniform
0: Gaussian normal
>0: super-gaussian(looks like triangle) laplace

Footnotes

Kevin Frans 2016, A[sic!] intuitive explanation of natural gradient descent ↩