Sparse PCA via Bipartite Matchings

Megasthenis Asteris; Dimitris Papailiopoulos; Anastasios Kyrillidis; Alexandros G. Dimakis

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arXiv · 1508.00625

Sparse PCA via Bipartite Matchings

Megasthenis Asteris·Dimitris Papailiopoulos·Anastasios Kyrillidis·Alexandros G. Dimakis

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Abstract

We consider the following multi-component sparse PCA problem: given a set of data points, we seek to extract a small number of sparse components with disjoint supports that jointly capture the maximum possible variance. These components can be computed one by one, repeatedly solving the single-component problem and deflating the input data matrix, but as we show this greedy procedure is suboptimal. We present a novel algorithm for sparse PCA that jointly optimizes multiple disjoint components. The extracted features capture variance that lies within a multiplicative factor arbitrarily close to 1 from the optimal. Our algorithm is combinatorial and computes the desired components by solving multiple instances of the bipartite maximum weight matching problem. Its complexity grows as a low order polynomial in the ambient dimension of the input data matrix, but exponentially in its rank. However, it can be effectively applied on a low-dimensional sketch of the data; this allows us to obtain polynomial-time approximation guarantees via spectral bounds. We evaluate our algorithm on real data-sets and empirically demonstrate that in many cases it outperforms existing, deflation-based approaches.

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BibTeX RIS

Megasthenis Asteris, Dimitris Papailiopoulos, Anastasios Kyrillidis, Alexandros G. Dimakis. 2015-08-04. Sparse PCA via Bipartite Matchings. https://arxiv.org/abs/1508.00625

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