large-Scale Supervised Sparse Principal Component Analysis

From statwiki
Revision as of 21:53, 4 August 2013 by Z47xu (talk | contribs) (1. Introduction)
Jump to: navigation, search

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]O(\hat{n^3})[/math], where [math]\hat{n}[/math] is the intrinsic dimension of the data. Since [math]\hat{n}[/math] could be very small compared to the dimension [math]n[/math] of the data, this algorithm is computationally easy.

2. Primal problem

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

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

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

[math]max_z[/math] Tr[math]\Sigma Z - \lambda \| Z \|_1 : Z \succeq 0[/math], Tr [math]Z=1[/math].

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