Using ?1-Relaxation and Integer Programming to Obtain Dual Bounds for Sparse PCA

Computational Lower Bounds for Sparse PCA

In the context of sparse principal component detection, we bring evidence towards the existence of a statistical price to pay for computational efficiency. We measure the performance of a test by the smallest signal strength that it can detect and we propose a computationally efficient method based on semidefinite programming. We also prove that the statistical performance of this test cannot b...

Lagrangian Relaxation for Integer Programming

It is a pleasure to write this commentary because it offers an opportunity to express my gratitude to several people who helped me in ways that turned out to be essential to the birth of [8]. They also had a good deal to do with shaping my early career and, consequently, much of what followed. The immediate event that triggered my interest in this topic occurred early in 1971 in connection with...

A Direct Formulation for Sparse PCA Using Semidefinite Programming

Given a covariance matrix, we consider the problem of maximizing the variance explained by a particular linear combination of the input variables while constraining the number of nonzero coefficients in this combination. This problem arises in the decomposition of a covariance matrix into sparse factors or sparse PCA, and has wide applications ranging from biology to finance. We use a modificat...

Spectral Bounds for Sparse PCA: Exact and Greedy Algorithms

Sparse PCA seeks approximate sparse “eigenvectors” whose projections capture the maximal variance of data. As a cardinality-constrained and non-convex optimization problem, it is NP-hard and is encountered in a wide range of applied fields, from bio-informatics to finance. Recent progress has focused mainly on continuous approximation and convex relaxation of the hard cardinality constraint. In...

Lower bounds for integer programming problems