large-Scale Supervised Sparse Principal Component Analysis

1. Introduction

The drawbacks of most existing technique:

1 Drawbacks of Existing techniques

Existing techniques include ad-hoc methods(e.g. factor rotation techniques, simple thresholding), greedy algorithms, SCoTLASS, the regularized SVD method, SPCA, the generalized power method. These methods are based on non-convex optimization and they don't guarantee global optimum.

A semi-definite relaxation method called DSPCA can guarantee global convergence and has better performance than above algorithms, however, it is computationally expensive.

2 Contribution of this paper

This paper solves DSPCA in a computationally easier way, and hence it is a good solution for large scale data sets. This paper applies a block coordinate ascent algorithm with computational complexity [math]\displaystyle{ O(\hat{n^3}) }[/math], where [math]\displaystyle{ \hat{n} }[/math] is the intrinsic dimension of the data. Since [math]\displaystyle{ \hat{n} }[/math] could be very small compared to the dimension [math]\displaystyle{ n }[/math] of the data, this algorithm is computationally easy.

2. Primal problem

The sparse PCA problem can be formulated as [math]\displaystyle{ max_x \ x^T \Sigma x - \lambda \| x \|_0 : \| x \|_2=1 }[/math].

This is equivalent to [math]\displaystyle{ max_z }[/math] Tr[math]\displaystyle{ \Sigma Z - \lambda \sqrt{\| Z \|_0} : Z \succeq 0 }[/math], Tr [math]\displaystyle{ Z=1 }[/math], Rank[math]\displaystyle{ (Z)=1 }[/math].

Replacing the [math]\displaystyle{ \sqrt{\| Z \|_0} }[/math] with [math]\displaystyle{ \| Z \|_1 }[/math] and dropping the rank constraint gives a relaxation of the original non-convex problem:

[math]\displaystyle{ \phi = max_z Tr (\Sigma Z) - \lambda \| Z \|_1 : Z \succeq 0 }[/math], [math]\displaystyle{ Tr(Z)=1 \qquad (1) }[/math] .

Fortunately, this relaxation approximates the original non-convex problem to a convex problem.

Here is an important theorem used by this paper:

Theorem(2.1) Let [math]\displaystyle{ \Sigma=A^T A }[/math] where [math]\displaystyle{ A=(a_1,a_2,......,a_n) \in {\mathbb R}^{m \times n} }[/math], we have [math]\displaystyle{ \psi = max_{\| \xi \|_2=1} }[/math] [math]\displaystyle{ \sum_{i=1}^{n} (({a_i}^T \xi)^2 - \lambda)_+ }[/math]. An optimal non-zero pattern corresponds to the indices [math]\displaystyle{ i }[/math] with [math]\displaystyle{ \lambda \lt (({a_i}^T \xi)^2-\lambda)_+ }[/math]

3. Block Coordinate Ascent Algorithm

There is a row-by-row algorithm applied to the problems of the form [math]\displaystyle{ min_X \ f(X)-\beta }[/math] log det [math]\displaystyle{ X: \ L \leq X \leq U, X }[/math]