Statistical and Computational Tradeoffs in High-dimensional Problems

Abstract

With the recent data revolution, statisticians are considering larger datasets, more sophisticated models, more complex problems. As a consequence, the algorithmic aspect of statistical methods can no longer be neglected in a world where computational power is the bottleneck, not the lack of observations. In this context, we present in this thesis results that establish fundamental limits in the statistical performance of computationally efficient procedures, for the problem of sparse principal component analysis. We will show how it is achieved through average-case reduction to the planted clique problem. We will also introduce further areas of research in this promising eld, related to the detection of planted satisability in boolean formulas.