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Improved SumofSquares Lower Bounds for Hidden Clique and Hidden Submatrix Problems
, 2015
"... Given a large data matrix A ∈ Rn×n, we consider the problem of determining whether its entries are i.i.d. with some known marginal distribution Aij ∼ P0, or instead A contains a principal submatrix AQ,Q whose entries have marginal distribution Aij ∼ P1 6 = P0. As a special case, the hidden (or plant ..."
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Given a large data matrix A ∈ Rn×n, we consider the problem of determining whether its entries are i.i.d. with some known marginal distribution Aij ∼ P0, or instead A contains a principal submatrix AQ,Q whose entries have marginal distribution Aij ∼ P1 6 = P0. As a special case, the hidden (or planted) clique problem requires to find a planted clique in an otherwise uniformly random graph. Assuming unbounded computational resources, this hypothesis testing problem is statistically solvable provided Q  ≥ C log n for a suitable constant C. However, despite substantial effort, no polynomial time algorithm is known that succeeds with high probability when Q  = o(√n). Recently Meka and Wigderson [MW13b], proposed a method to establish lower bounds within the Sum of Squares (SOS) semidefinite hierarchy. Here we consider the degree4 SOS relaxation, and study the construction of [MW13b] to prove that SOS fails unless k ≥ C n1/3 / log n. An argument presented by Barak implies that this lower bound cannot be substantially improved unless the witness construction is changed in the proof. Our proof uses the moments method to bound the spectrum of a certain random
Statistical Limits of Convex Relaxations
"... Many high dimensional sparse learning problems are formulated as nonconvex optimization. A popular approach to solve these nonconvex optimization problems is through convex relaxations such as linear and semidefinite programming. In this paper, we study the statistical limits of convex relaxations. ..."
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Many high dimensional sparse learning problems are formulated as nonconvex optimization. A popular approach to solve these nonconvex optimization problems is through convex relaxations such as linear and semidefinite programming. In this paper, we study the statistical limits of convex relaxations. Particularly, we consider two problems: Mean estimation for sparse principal submatrix and edge probability estimation for stochastic block model. We exploit the sumofsquares relaxation hierarchy to sharply characterize the limits of a broad class of convex relaxations. Our result shows statistical optimality needs to be compromised for achieving computational tractability using convex relaxations. Compared with existing results on computational lower bounds for statistical problems, which consider general polynomialtime algorithms and rely on computational hardness hypotheses on problems like planted clique detection, our theory focuses on a broad class of convex relaxations and does not rely on unproven hypotheses. 1
How to refute a random CSP
, 2015
"... Let P be a nontrivial kary predicate over a finite alphabet. Consider a random CSP(P) instance I over n variables with m constraints, each being P applied to k random literals. When m n the instance I will be unsatisfiable with high probability, and the natural associated algorithmic task is to fi ..."
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Let P be a nontrivial kary predicate over a finite alphabet. Consider a random CSP(P) instance I over n variables with m constraints, each being P applied to k random literals. When m n the instance I will be unsatisfiable with high probability, and the natural associated algorithmic task is to find a refutation of I — i.e., a certificate of unsatisfiability. When P is the 3ary Boolean OR predicate, this is the well studied problem of refuting random 3SAT formulas; in this case, an efficient algorithm is known only when m n3/2. Understanding the density required for averagecase refutation of other predicates is of importance for various areas of complexity, including cryptography, proof complexity, and learning theory. The main previouslyknown result is that for a general Boolean kary predicate P, having m ndk/2e random constraints suffices for efficient refutation. In this work we give a general criterion for arbitrary kary predicates, one that often yields efficient refutation algorithms at much lower densities. Specifically, if P fails to support a twise independent (uniform) probability distribution (2 ≤ t ≤ k), then there is an efficient algorithm that refutes random CSP(P) instances I with high probability, provided m nt/2. Indeed,
Improved SumofSquares Lower Bounds for Hidden Clique and Hidden Submatrix Problems
, 2015
"... Given a large data matrix A ∈ Rn×n, we consider the problem of determining whether its entries are i.i.d. with some known marginal distribution Aij ∼ P0, or instead A contains a principal submatrix AQ,Q whose entries have marginal distribution Aij ∼ P1 6 = P0. As a special case, the hidden (or plant ..."
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Given a large data matrix A ∈ Rn×n, we consider the problem of determining whether its entries are i.i.d. with some known marginal distribution Aij ∼ P0, or instead A contains a principal submatrix AQ,Q whose entries have marginal distribution Aij ∼ P1 6 = P0. As a special case, the hidden (or planted) clique problem requires to find a planted clique in an otherwise uniformly random graph. Assuming unbounded computational resources, this hypothesis testing problem is statistically solvable provided Q  ≥ C log n for a suitable constant C. However, despite substantial effort, no polynomial time algorithm is known that succeeds with high probability when Q  = o(√n). Recently Meka and Wigderson [MW13b], proposed a method to establish lower bounds within the Sum of Squares (SOS) semidefinite hierarchy. Here we consider the degree4 SOS relaxation, and study the construction of [MW13b] to prove that SOS fails unless k ≥ C n1/3 / log n. An argument presented by Barak implies that this lower bound cannot be substantially improved unless the witness construction is changed in the proof. Our proof uses the moments method to bound the spectrum of a certain random
Hardness of Approximation
"... Abstract. This article accompanies the talk given by the author at the International Congress of Mathematicians, 2014. The article sketches some connections between approximability of NPcomplete problems, analysis and geometry, and the role played by the Unique Games Conjecture in facilitating the ..."
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Abstract. This article accompanies the talk given by the author at the International Congress of Mathematicians, 2014. The article sketches some connections between approximability of NPcomplete problems, analysis and geometry, and the role played by the Unique Games Conjecture in facilitating these connections. For a more extensive introduction to the topic, the reader is referred to survey articles [39, 40, 64].