We show the existence of rigid combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, t-designs, and t-wise permutations. In all cases, the sizes of the objects ar e optimal up to polynomial overhead. The proof of existence is probabilistic. We show that a randomly chosen such object has the required properties with positive yet tiny pr obability. The main technical ingredient is a special local central limit theorem for suitable lattice ra ndom walks with finitely many steps.
One relaxation is to allow a set @math with a non-uniform distribution @math . For many practical applications of @math -designs and @math -wise permutations in statistics and computer science, but not quite every application, this relaxation is as good as the uniform question. The existence of a solution with small support is guaranteed by Carath 'eodory's theorem, using the fact that the constraints on @math are all linear equalities and inequalities. Moreover, such a solution can be found efficiently, as was shown by Karp and Papadimitriou @cite_2 and in more general settings by Koller and Megiddo @cite_6 . Alon and Lovett @cite_3 give a strongly explicit analog of this in the case of @math -wise permutations and more generally in the case of group actions.
@cite_3 @cite_6 @cite_2
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Lov asz and Schrijver [SIAM J. Optim., 1 (1991), pp. 166--190] devised a lift-and-project method that produces a sequence of convex relaxations for the problem of finding in a graph an independent set (or a clique) of maximum size. Each relaxation in the sequence is tighter than the one before it, while the first relaxation is already at least as strong as the Lov asz theta function [IEEE Trans. Inform. Theory, 25 (1979), pp. 1--7]. We show that on a random graph Gn,1 2, the value of the rth relaxation in the sequence is roughly 0pt 7pt @math , almost surely. It follows that for those relaxations known to be efficiently computable, namely, for r=O(1), the value of the relaxation is comparable to the theta function. Furthermore, a perfectly tight relaxation is almost surely obtained only at the @math relaxation in the sequence. Let B i be deterministic real symmetric m × m matrices, and ξ i be independent random scalars with zero mean and “of order of one” (e.g., @math ). We are interested to know under what conditions “typical norm” of the random matrix @math is of order of 1. An evident necessary condition is @math , which, essentially, translates to @math ; a natural conjecture is that the latter condition is sufficient as well. In the paper, we prove a relaxed version of this conjecture, specifically, that under the above condition the typical norm of S N is @math : @math for all Ω > 0 We outline some applications of this result, primarily in investigating the quality of semidefinite relaxations of a general quadratic optimization problem with orthogonality constraints @math , where F is quadratic in X = (X 1,... ,X k ). We show that when F is convex in every one of X j , a natural semidefinite relaxation of the problem is tight within a factor slowly growing with the size m of the matrices @math . We study exact recovery conditions for convex relaxations of point cloud clustering problems, focusing on two of the most common optimization problems for unsupervised clustering: @math -means and @math -median clustering. Motivations for focusing on convex relaxations are: (a) they come with a certificate of optimality, and (b) they are generic tools which are relatively parameter-free, not tailored to specific assumptions over the input. More precisely, we consider the distributional setting where there are @math clusters in @math and data from each cluster consists of @math points sampled from a symmetric distribution within a ball of unit radius. We ask: what is the minimal separation distance between cluster centers needed for convex relaxations to exactly recover these @math clusters as the optimal integral solution? For the @math -median linear programming relaxation we show a tight bound: exact recovery is obtained given arbitrarily small pairwise separation @math between the balls. In other words, the pairwise center separation is @math . Under the same distributional model, the @math -means LP relaxation fails to recover such clusters at separation as large as @math . Yet, if we enforce PSD constraints on the @math -means LP, we get exact cluster recovery at center separation @math . In contrast, common heuristics such as Lloyd's algorithm (a.k.a. the @math -means algorithm) can fail to recover clusters in this setting; even with arbitrarily large cluster separation, k-means++ with overseeding by any constant factor fails with high probability at exact cluster recovery. To complement the theoretical analysis, we provide an experimental study of the recovery guarantees for these various methods, and discuss several open problems which these experiments suggest. Let @math be a nontrivial @math -ary predicate. Consider a random instance of the constraint satisfaction problem @math on @math variables with @math constraints, each being @math applied to @math randomly chosen literals. Provided the constraint density satisfies @math , such an instance is unsatisfiable with high probability. The problem is to efficiently find a proof of unsatisfiability. We show that whenever the predicate @math supports a @math - probability distribution on its satisfying assignments, the sum of squares (SOS) algorithm of degree @math (which runs in time @math ) refute a random instance of @math . In particular, the polynomial-time SOS algorithm requires @math constraints to refute random instances of CSP @math when @math supports a @math -wise uniform distribution on its satisfying assignments. Together with recent work of [LRS15], our result also implies that polynomial-size semidefinite programming relaxation for refutation requires at least @math constraints. Our results (which also extend with no change to CSPs over larger alphabets) subsume all previously known lower bounds for semialgebraic refutation of random CSPs. For every constraint predicate @math , they give a three-way hardness tradeoff between the density of constraints, the SOS degree (hence running time), and the strength of the refutation. By recent algorithmic results of [AOW15] and [RRS16], this full three-way tradeoff is , up to lower-order factors.