Sparse Principal Component Analysis via the Sparsest Element in a Subspace

点击翻译按钮获取中文摘要

Sparse principal component analysis (PCA) is a technique that allows biologists and other scientists to interpret experimental data in terms of very few variables. For example, it can help identify which among thousands of genes are important in distinguishing different types of cancer. In order for scientists and engineers to select the best algorithm for finding sparse principal components, it is important to have a theoretical understanding of the performance of many algorithms under a realistic data model. Most existing theoretical understanding focuses on the simple case where there is a single component that happens to be sparse. The proposed work will introduce a new model in which there are multiple components, of which one is sparse. For a special case of this more realistic model, the proposed work attempts to understand if there are any conditions under which sophisticated convex programs are provably better than very simple algorithms. Either outcome would be informative in helping researchers decide between the many algorithms for sparse PCA. In this project, sparse PCA will be studied from the perspective of finding the sparsest element in a subspace. This perspective is motivated by a multispike data model, which the PI calls a sparse-dense model. Under this model, the infinite data limit of sparse PCA becomes the sparsest element problem, which is nontrivial. The objective of this research is to understand the computational-statistical tradeoff in finding the sparsest element in a subspace under the sparse-dense model. The PI would like to determine if there is a scaling gap between the information theoretic limit and the best performance by a computationally efficient algorithm. Ultimately, we would like to understand when sophisticated convex methods are provably better than simple thresholding methods. This objective will be explored by semidefinite relaxations, polynomial optimization, and reductions to the planted clique problem.