Clustering is a fundamental primitive in unsupervised learning which gives rise to a rich class
of computationally-challenging inference tasks. In this work, we focus on the canonical task of
clustering d-dimensional Gaussian mixtures with unknown (and possibly degenerate) covariance.
Recent works (Ghosh et al. ’20; Mao, Wein ’21; Davis, Diaz, Wang ’21) have established lower
bounds against the class of low-degree polynomial methods and the sum-of-squares (SoS) hierarchy
for recovering certain hidden structures planted in Gaussian clustering instances. Prior work
on many similar inference tasks portends that such lower bounds strongly suggest the presence
of an inherent statistical-to-computational gap for clustering, that is, a parameter regime where
the clustering task is statistically possible but no polynomial-time algorithm succeeds.
One special case of the clustering task we consider is equivalent to the problem of finding a
planted hypercube vector in an otherwise random subspace. We show that, perhaps surprisingly,
this particular clustering model does not exhibit a statistical-to-computational gap, despite the
aforementioned low-degree and SoS lower bounds. To achieve this, we give an algorithm based
on Lenstra–Lenstra–Lovász lattice basis reduction which achieves the statistically-optimal sample
complexity of d + 1 samples. This result extends the class of problems whose conjectured
statistical-to-computational gaps can be “closed” by “brittle” polynomial-time algorithms, highlighting
the crucial but subtle role of noise in the onset of statistical-to-computational gaps.
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Free Energy Wells and Overlap Gap Property in Sparse PCA
We study a variant of the sparse PCA (principal component analysis) problem in the “hard” regime, where the inference task is possible yet no polynomial-time algorithm is known to exist. Prior work, based on the low-degree likelihood ratio, has conjectured a precise expression for the best possible (sub-exponential) runtime throughout the hard regime. Following instead a statistical physics inspired point of view, we show bounds on the depth of free energy wells for various Gibbs measures naturally associated to the problem. These free energy wells imply hitting time lower bounds that corroborate the low-degree conjecture: we show that a class of natural MCMC (Markov chain Monte Carlo) methods (with worst-case initialization) cannot solve sparse PCA with less than the conjectured runtime. These lower bounds apply to a wide range of values for two tuning parameters: temperature and sparsity misparametrization. Finally, we prove that the Overlap Gap Property (OGP), a structural property that implies failure of certain local search algorithms, holds in a significant part of the hard regime.
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- Award ID(s):
- 1712730
- NSF-PAR ID:
- 10259973
- Editor(s):
- Abernethy, Jacob; Agarwal, Shivani
- Date Published:
- Journal Name:
- Proceedings of Thirty Third Conference on Learning Theory
- Volume:
- 125
- Page Range / eLocation ID:
- 479 - 482
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
