Full regularization path for sparse principal component analysis

Given a sample covariance matrix, we examine 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 is known as sparse principal component analysis and has a wide array of applications in machine learning and engineering. We formulate a new semidefinite relaxation to this problem and derive a greedy algorithm that computes a full set of good solutions for all numbers of non zero coefficients, with complexity O(n3 ), where n is the number of variables. We then use the same relaxation to derive sufficient conditions for global optimality of a solution, which can be tested in O(n3 ). We show on toy examples and biological data that our algorithm does provide globally optimal solutions in many cases.

• Type: Conference
• Year: 2007
• Where: ICML
• Authors: Alexandre d'Aspremont, Francis R. Bach, Laurent El Ghaoui
• Keywords: ICML 2007, Machine Learning, Non Zero Coefficients, Nonzero Coefficients, Particular Linear Combination