
Computational Hardness of Certifying Bounds on Constrained PCA Problems
Given a random n × n symmetric matrix W drawn from the Gaussian orthogo...
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Local Statistics, Semidefinite Programming, and Community Detection
We propose a new hierarchy of semidefinite programming relaxations for i...
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Even flying cops should think ahead
We study the entanglement game, which is a version of cops and robbers, ...
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Notes on Computational Hardness of Hypothesis Testing: Predictions using the LowDegree Likelihood Ratio
These notes survey and explore an emerging method, which we call the low...
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Classical algorithms and quantum limitations for maximum cut on highgirth graphs
We study the performance of local quantum algorithms such as the Quantum...
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Large independent sets on random dregular graphs with d small
In this paper, we present a prioritized local algorithm that computes a ...
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Joint estimation of parameters in Ising model
We study joint estimation of the inverse temperature and magnetization p...
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Spectral Planting and the Hardness of Refuting Cuts, Colorability, and Communities in Random Graphs
We study the problem of efficiently refuting the kcolorability of a graph, or equivalently certifying a lower bound on its chromatic number. We give formal evidence of averagecase computational hardness for this problem in sparse random regular graphs, showing optimality of a simple spectral certificate. This evidence takes the form of a computationallyquiet planting: we construct a distribution of dregular graphs that has significantly smaller chromatic number than a typical regular graph drawn uniformly at random, while providing evidence that these two distributions are indistinguishable by a large class of algorithms. We generalize our results to the more general problem of certifying an upper bound on the maximum kcut. This quiet planting is achieved by minimizing the effect of the planted structure (e.g. colorings or cuts) on the graph spectrum. Specifically, the planted structure corresponds exactly to eigenvectors of the adjacency matrix. This avoids the pushout effect of random matrix theory, and delays the point at which the planting becomes visible in the spectrum or local statistics. To illustrate this further, we give similar results for a Gaussian analogue of this problem: a quiet version of the spiked model, where we plant an eigenspace rather than adding a generic lowrank perturbation. Our evidence for computational hardness of distinguishing two distributions is based on three different heuristics: stability of belief propagation, the local statistics hierarchy, and the lowdegree likelihood ratio. Of independent interest, our results include generalpurpose bounds on the lowdegree likelihood ratio for multispiked matrix models, and an improved lowdegree analysis of the stochastic block model.
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