Approximating Sparse PCA from Incomplete Data

TL;DR

This paper presents a method to approximate sparse principal components from incomplete data using spectral norm sketches, achieving near-optimal solutions efficiently across various data types.

Contribution

It introduces a spectral norm sketching approach for sparse PCA that reduces computational complexity while maintaining approximation quality.

Findings

Achieves psilon}-additive approximation with alculated number of elements

Reduces running time by a factor of five or more

Effective across image, text, biological, and financial data

Abstract

We study how well one can recover sparse principal components of a data matrix using a sketch formed from a few of its elements. We show that for a wide class of optimization problems, if the sketch is close (in the spectral norm) to the original data matrix, then one can recover a near optimal solution to the optimization problem by using the sketch. In particular, we use this approach to obtain sparse principal components and show that for \math{m} data points in \math{n} dimensions, \math{O(\epsilon^{-2}\tilde k\max\{m,n\})} elements gives an \math{\epsilon}-additive approximation to the sparse PCA problem (\math{\tilde k} is the stable rank of the data matrix). We demonstrate our algorithms extensively on image, text, biological and financial data. The results show that not only are we able to recover the sparse PCAs from the incomplete data, but by using our sparse sketch, the running time drops by a factor of five or more.