Hidden Integrality and Semi-random Robustness of SDP Relaxation for Sub-Gaussian Mixture Model

TL;DR

This paper proves that SDP relaxations for Sub-Gaussian Mixture Models have a hidden integrality property, with exponentially decaying error bounds, and are robust under semi-random modifications, enabling near-exact clustering in challenging regimes.

Contribution

The paper establishes a hidden integrality property of SDP relaxations for mixture models, with exponential error decay and semi-random robustness, advancing theoretical understanding of clustering algorithms.

Findings

Error bounds decay exponentially with signal-to-noise ratio.

SDP solutions are integral and exact in certain regimes.

SDP achieves near-perfect recovery under the stochastic ball model.

Abstract

We consider the problem of estimating the discrete clustering structures under the Sub-Gaussian Mixture Model. Our main results establish a hidden integrality property of a semidefinite programming (SDP) relaxation for this problem: while the optimal solution to the SDP is not integer-valued in general, its estimation error can be upper bounded by that of an idealized integer program. The error of the integer program, and hence that of the SDP, are further shown to decay exponentially in the signal-to-noise ratio. In addition, we show that the SDP relaxation is robust under the semi-random setting in which an adversary can modify the data generated from the mixture model. In particular, we generalize the hidden integrality property to the semi-random model and thereby show that SDP achieves the optimal error bound in this setting. These results together highlight the "global-to-local" mechanism that drives the performance of the SDP relaxation. To the best of our knowledge, our result is the first exponentially decaying error bound for convex relaxations of mixture models. A corollary of our results shows that in certain regimes the SDP solutions are in fact integral and exact. More generally, our results establish sufficient conditions for the SDP to correctly recover the cluster memberships of $(1-\delta)$ fraction of the points for any $\delta\in(0,1)$. As a special case, we show that under the $d$-dimensional Stochastic Ball Model, SDP achieves non-trivial (sometimes exact) recovery when the center separation is as small as $\sqrt{1/d}$, which improves upon previous exact recovery results that require constant separation.