Finding Hidden Cliques in Linear Time with High Probability

This paper studies the minimax detection of a small submatrix of elevated mean in a large matrix contaminated by additive Gaussian noise. To investigate the tradeoff between statistical performance and computational cost from a complexity-theoretic perspective, we consider a sequence of discretized models which are asymptotically equivalent to the Gaussian model. Under the hypothesis that the planted clique detection problem cannot be solved in randomized polynomial time when the clique size is of smaller order than the square root of the graph size, the following phase transition phenomenon is established: when the size of the large matrix p → ∞, if the submatrix size k = Θ(p α ) for any α ∈ (0, 2/3), computational complexity constraints can incur a severe penalty on the statistical performance in the sense that any randomized polynomial-time test is minimax suboptimal by a polynomial factor in p; if k = Θ(p α ) for any α ∈ (2/3, 1), minimax optimal detection can be attained within constant factors in linear time. Using Schatten norm loss as a representative example, we show that the hardness of attaining the minimax estimation rate can crucially depend on the loss function. Implications on the hardness of support recovery are also obtained.

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“…The randomized mapping from A to X is as follows. By (18), N is even, and let N 2 = N/2 = pℓ and [N ] \ [N 2 ] = {N 2 + 1, . .…”

Section: 1mentioning

confidence: 99%

Computational barriers in minimax submatrix detection

Ma¹,

Wu²

2015

Ann. Statist.

110

158

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“…[6] and [7] for the general framework as well as application areas motivating such questions and see [2] and [12] for a number of interesting general results in these contexts. In the context of the combinatorics, such questions result in the famous planted clique problem see e.g [5,15] and the references therein.…”

Section: Loopmentioning

confidence: 99%

Energy landscape for large average submatrix detection problems in Gaussian random matrices

Bhamidi

Dey

Nobel

2017

Probab. Theory Relat. Fields

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The problem of finding large average submatrices of a real-valued matrix arises in the exploratory analysis of data from a variety of disciplines, ranging from genomics to social sciences. In this paper we provide a detailed asymptotic analysis of large average submatrices of an n × n Gaussian random matrix. The first part of the paper addresses global maxima. For fixed k we identify the average and the joint distribution of the k × k submatrix having largest average value. As a dual result, we establish that the size of the largest square sub-matrix with average bigger than a fixed positive constant is, with high probability, equal to one of two consecutive integers that depend on the threshold and the matrix dimension n. The second part of the paper addresses local maxima. Specifically we consider submatrices with dominant row and column sums that arise as the local optima of iterative search procedures for large average submatrices. For fixed k, we identify the limiting average value and joint distribution of a k × k submatrix conditioned to be a local maxima. In order to understand the density of such local optima and explain the quick convergence of such iterative procedures, we analyze the number Ln(k) of local maxima, beginning with exact asymptotic expressions for the mean and fluctuation behavior of Ln(k). For fixed k, the mean of Ln(k) is Θ(n k /(log n) (k−1)/2 ) while the standard deviation is Θ(n 2k 2 /(k+1) /(log n) k 2 /(k+1) ). Our principal result is a Gaussian central limit theorem for Ln(k) that is based on a new variant of Stein's method.

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“…Similarly, the subdifferential of the nuclear norm atX is equal to the set ∂ X * = {vv T /k + W : Wv = W Tv = 0, W ≤ 1}; see [48,Example 2]. Combining (36) and (37) and substituting this formula for the subgradients of X * into the resulting equation yields (18) and (19). Thus, the conditions (18), (19), (20), (21), and (22) are exactly the Karush-Kuhn-Tucker conditions for (6) applied at (X,Ȳ ), with the Lagrange multiplier M 2 taken to be 0.…”

Section: A Appendicesmentioning

confidence: 99%

Guaranteed Recovery of Planted Cliques and Dense Subgraphs by Convex Relaxation

Ames

2015

J Optim Theory Appl

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We consider the problem of identifying the densest k-node subgraph in a given graph. We write this problem as an instance of rank-constrained cardinality minimization and then relax using the nuclear and 1 norms. Although the original combinatorial problem is NP-hard, we show that the densest k-subgraph can be recovered from the solution of our convex relaxation for certain program inputs. In particular, we establish exact recovery in the case that the input graph contains a single planted clique plus noise in the form of corrupted adjacency relationships. We consider two constructions for this noise. In the first, noise is introduced by an adversary deterministically deleting edges within the planted clique and placing diversionary edges. In the second, these edge corruptions are performed at random. Analogous recovery guarantees for identifying the densest subgraph of fixed size in a bipartite graph are also established, and results of numerical simulations for randomly generated graphs are included to demonstrate the efficacy of our algorithm.

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Finding Hidden Cliques in Linear Time with High Probability

Cited by 50 publications

References 24 publications

Computational barriers in minimax submatrix detection

Computational barriers in minimax submatrix detection

Energy landscape for large average submatrix detection problems in Gaussian random matrices

Guaranteed Recovery of Planted Cliques and Dense Subgraphs by Convex Relaxation

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