Computationally efficient sparse clustering

Abstract

We study statistical and computational limits of clustering when the means of the centres are sparse and their dimension is possibly much larger than the sample size. Our theoretical analysis focuses on the simple model, which has two clusters with centres θ and −θ.We provide a finite sample analysis of a new sparse clustering algorithm based on sparse PCA and show that it achieves the minimax optimal misclustering rate in the regime θ → ∞, matching asymptotically the Bayes error.Our results require the sparsity to grow slower than the square root of the sample size. Using a recent framework for computational lower boundsthe low-degree likelihood ratio-we give evidence that this condition is necessary for any polynomial-time clustering algorithm to succeed below the BBP threshold. This complements existing evidence based on reductions and statistical query lower bounds. Compared to these existing results, we cover a wider set of parameter regimes and give a more precise understanding of the runtime required and the misclustering error achievable.We also discuss extensions of our results to more than two clusters.MSC 2010 subject classifications: 62H30.

High-dimensional logistic entropy clustering

“…This framework is indeed a good benchmark, since on the one hand, it is sufficiently simple to allow us to understand the nature of the target (β 0,1 , ..., β 0,K ) -with K = 2 and β 0,1 = −β 0,2 in our bipartite framework -and to investigate the rate of convergence of estimators of the form of (2), suitably regularized by a 1 -penalty. On the other hand, the two-component high-dimensional Gaussian mixture has received recently at lot of attention [7,2,35,28,24,15,12,3,25,8,30]. Let us emphasize that our goal is not a priori to provide a state-of-the-art method, specifically designed to solve the high-dimensional…”