Independent Component Analysis

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Introduction

If you recall, in sparse coding, we wanted to learn an over-complete basis for the data. In particular, this implies that the basis vectors that we learn in sparse coding will not be linearly independent. While this may be desirable in certain situations, sometimes we want to learn a linearly independent basis for the data. In independent component analysis (ICA), this is exactly what we want to do. Further, in ICA, we want to learn not just any linearly independent basis, but an orthonormal basis for the data. (An orthonormal basis is a basis (\phi_1, \ldots \phi_n) such that \phi_i \cdot \phi_j = 0 if i \ne j and 1 if i = j).

Like sparse coding, independent component analysis has a simple mathematical formulation. Given some data x, we would like to learn a set of basis vectors which we represent in the columns of a matrix W, such that, firstly, as in sparse coding, our features are sparse; and secondly, our basis is an orthonormal basis. (Note that while in sparse coding, our matrix A was for mapping features s to raw data, in independent component analysis, our matrix W works in the opposite direction, mapping raw data x to features instead). This gives us the following objective function:


J(W) = \lVert Wx \rVert_1

This objective function is equivalent to the sparsity penalty on the features s in sparse coding, since Wx is precisely the features that represent the data. Adding in the orthonormality constraint gives us the full optimization problem for independent component analysis:


\begin{array}{rcl}
     {\rm minimize} & \lVert Wx \rVert_1  \\
     {\rm s.t.}     & WW^T = I \\
\end{array}

As is usually the case in deep learning, this problem has no simple analytic solution, and to make matters worse, the orthonormality constraint makes it slightly more difficult to optimize for the objective using gradient descent - every iteration of gradient descent must be followed by a step that maps the new basis back to the space of orthonormal bases (hence enforcing the constraint).

In practice, optimizing for the objective function while enforcing the orthonormality constraint (as described in Orthonormal ICA section below) is feasible but slow. Hence, the use of orthonormal ICA is limited to situations where it is important to obtain an orthonormal basis (TODO: what situations) . In other situations, the orthonormality constraint is replaced by a reconstruction cost instead to give Reconstruction ICA, which is very similar to sparse coding, but no longer strictly finds independent components.

Orthonormal ICA

The orthonormal ICA objective is:


\begin{array}{rcl}
     {\rm minimize} & \lVert Wx \rVert_1  \\
     {\rm s.t.}     & WW^T = I \\
\end{array}

Observe that the constraint WWT = I implies two other constraints.

Firstly, since we are learning an orthonormal basis, the number of basis vectors we learn must be less than the dimension of the input. In particular, this means that we cannot learn over-complete bases as we usually do in sparse coding.

Secondly, the data must be ZCA whitened with no regularization (that is, with ε set to 0). (TODO Why must this be so?)

Hence, before we even begin to optimize for the orthonormal ICA objective, we must ensure that our data has been whitened, and that we are learning an under-complete basis.

Following that, to optimize for the objective, we can use gradient descent, interspersing gradient descent steps with projection steps to enforce the orthonormality constraint. Hence, the procedure will be as follows:

Repeat until done:

  1. W \leftarrow W - \alpha \nabla_W \lVert Wx \rVert_1
  2. W \leftarrow \operatorname{proj}_U W where U is the space of matrices satisfying WWT = I

In practice, the learning rate α is varied using a line-search algorithm to speed up the descent, and the projection step is achieved by setting W \leftarrow (WW^T)^{-\frac{1}{2}} W, which can actually be seen as ZCA whitening (TODO explain how it is like ZCA whitening).

Reconstruction ICA

In reconstruction ICA, we drop the constraint that we want an orthonormal basis, replacing it with a reconstruction error term. Hence, the new reconstruction ICA objective is:


\begin{array}{rcl}
     {\rm minimize} & \lVert Wx \rVert_1 + \lVert W^TWx \rVert_2^2 \\
\end{array}

where WT(Wx) is the reconstruction of the input from the features (i.e. WT is supposed to play the role of W − 1). If you recall the sparse coding objective,


J(A, s) = \lVert As - x \rVert_2^2 + \lambda \lVert s \rVert_1

you will notice that these two objectives are identical, except that whereas in sparse coding, we are trying to learn a matrix A that maps the feature space to the input space, in reconstruction ICA, we are trying to learn a matrix W that maps the input space to the feature space instead.

As such, it is obvious that reconstruction ICA does not learn independent or orthonormal components.

Topographic ICA

Just like sparse coding, independent component analysis can be modified to give a topographic variant by adding a topographic cost term.

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