Tselil Schramm

Fast spectral algorithms from sum-of-squares proofs: tensor decomposition and planted sparse vectors

We consider two problems that arise in machine learning applications: the problem of recovering a planted sparse vector in a random linear subspace and the problem of decomposing a random low-rank overcomplete 3-tensor. For both problems, the best known guarantees are based on the sum-of-squares method. We develop new algorithms inspired by analyses of the sum-of-squares method. Our algorithms achieve the same or similar guarantees as sum-of-squares for these problems but the running time is significantly faster. For the planted sparse vector problem, we give an algorithm with running time nearly linear in the input size that approximately recovers a planted sparse vector with up to constant relative sparsity in a random subspace of ℝn of dimension up to Ω(√n). These recovery guarantees match the best known ones of Barak, Kelner, and Steurer (STOC 2014) up to logarithmic factors. For tensor decomposition, we give an algorithm with running time close to linear in the input size (with exponent ≈ 1.125) that approximately recovers a component of a random 3-tensor over ℝn of rank up to Ω(n4/3). The best previous algorithm for this problem due to Ge and Ma (RANDOM 2015) works up to rank Ω(n3/2) but requires quasipolynomial time.

Strongly refuting random CSPs below the spectral threshold

Random constraint satisfaction problems (CSPs) are known to exhibit threshold phenomena: given a uniformly random instance of a CSP with n variables and m clauses, there is a value of m = Ω(n) beyond which the CSP will be unsatisfiable with high probability. Strong refutation is the problem of certifying that no variable assignment satisfies more than a constant fraction of clauses; this is the natural algorithmic problem in the unsatisfiable regime (when m/n = ω(1)). Intuitively, strong refutation should become easier as the clause density m/n grows, because the contradictions introduced by the random clauses become more locally apparent. For CSPs such as k-SAT and k-XOR, there is a long-standing gap between the clause density at which efficient strong refutation algorithms are known, m/n ≥ Ο(nk/2-1), and the clause density at which instances become unsatisfiable with high probability, m/n = ω (1). In this paper, we give spectral and sum-of-squares algorithms for strongly refuting random k-XOR instances with clause density m/n ≥ Ο(n(k/2-1)(1-δ)) in time exp(Ο(nδ)) or in Ο(nδ) rounds of the sum-of-squares hierarchy, for any δ ∈ [0,1) and any integer k ≥ 3. Our algorithms provide a smooth transition between the clause density at which polynomial-time algorithms are known at δ = 0, and brute-force refutation at the satisfiability threshold when δ = 1. We also leverage our k-XOR results to obtain strong refutation algorithms for SAT (or any other Boolean CSP) at similar clause densities. Our algorithms match the known sum-of-squares lower bounds due to Grigoriev and Schonebeck, up to logarithmic factors.

Global and Local Information in Clustering Labeled Block Models

The stochastic block model is a classical cluster-exhibiting random graph model that has been widely studied in statistics, physics, and computer science. In its simplest form, the model is a random graph with two equal-sized clusters, with intracluster edge probability

Braess's paradox for the spectral gap in random graphs and delocalization of eigenvectors

p

Gap Amplification for Small-Set Expansion via Random Walks

, and intercluster edge probability

Computing Exact Minimum Cuts Without Knowing the Graph

q

On the Integrality Gap of Degree-4 Sum of Squares for Planted Clique

. We focus on the sparse case, i.e.,

Near Optimal LP Rounding Algorithm for CorrelationClustering on Complete and Complete k-partite Graphs

p, q = O(1/n)

Tight Lower Bounds for Planted Clique in the Degree-4 SOS Program

, which is practically more relevant and also mathematically more challenging. A conjecture of Decelle, Krzakala, Moore, and Zdeborova, based on ideas from statistical physics, predicted a specific threshold for clustering. The negative direction of the conjecture was proved by Mossel, Neeman, and Sly (2012), and more recently, the positive direction was independently proved by Massoulie and Mossel, Neeman, and Sly. In many real network clustering problems, nodes contain information as well. We study the interplay between node and network information in clustering by studying a labeled block model, where in addition to the edge information, the true cluster labels of a small fraction of the nodes are revealed. In the case of two clusters, we show that below the threshold, a small amount of node information does not affect recovery. On the other hand, we show that for any small amount of information, efficient local clustering is achievable as long as the number of clusters is sufficiently large (as a function of the amount of revealed information).

On the integrality gap of degree-4 sum of squares for planted clique

We study how the spectral gap of the normalized Laplacian of a random graph changes when an edge is added to or removed from the graph. There are known examples of graphs where, perhaps counter-intuitively, adding an edge can decrease the spectral gap, a phenomenon that is analogous to Braesss paradox in traffic networks. We show that this is often the case in random graphs in a strong sense. More precisely, we show that for typical instances of Erdi¾?s-Renyi random graphs Gn, p with constant edge density p∈0,1, the addition of a random edge will decrease the spectral gap with positive probability, strictly bounded away from zero. To do this, we prove a new delocalization result for eigenvectors of the Laplacian of Gn, p, which might be of independent interest.