text: Consider an Erd "os-Renyi random graph in which each edge is present independently with probability 1 2, except for a subset @math of the vertices that form a clique (a completely connected subgraph). We consider the problem of identifying the clique, given a realization of such a random graph. The best known algorithm provably finds the clique in linear time with high probability, provided @math dekel2011finding . Spectral methods can be shown to fail on cliques smaller than @math . In this paper we describe a nearly linear time algorithm that succeeds with high probability for @math for any @math . This is the first algorithm that provably improves over spectral methods. We further generalize the hidden clique problem to other background graphs (the standard case corresponding to the complete graph on @math vertices). For large girth regular graphs of degree @math we prove that local' algorithms succeed if @math and fail if @math .
  @cite_38: 
  @cite_22: The search for sample-variable associations is an important problem in the exploratory analysis of high dimensional data. Biclustering methods search for sample-variable associations in the form of distinguished submatrices of the data matrix. (The rows and columns of a submatrix need not be contiguous.) In this paper we propose and evaluate a statistically motivated biclustering procedure (LAS) that finds large average submatrices within a given real-valued data matrix. The procedure operates in an iterative-residual fashion, and is driven by a Bonferroni-based significance score that effectively trades off between submatrix size and average value. We examine the performance and potential utility of LAS, and compare it with a number of existing methods, through an extensive three-part validation study using two gene expression datasets. The validation study examines quantitative properties of biclusters, biological and clinical assessments using auxiliary information, and classification of disease subtypes using bicluster membership. In addition, we carry out a simulation study to assess the effectiveness and noise sensitivity of the LAS search procedure. These results suggest that LAS is an effective exploratory tool for the discovery of biologically relevant structures in high dimensional data. Software is available at https: genome.unc.edu las .
  @cite_0: 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 @math Gaussian random matrix. The first part of the paper addresses global maxima. For fixed @math we identify the average and the joint distribution of the @math 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 @math . 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 @math , we identify the limiting average value and joint distribution of a @math 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 @math of local maxima, beginning with exact asymptotic expressions for the mean and fluctuation behavior of @math . For fixed @math , the mean of @math is @math while the standard deviation is @math . Our principal result is a Gaussian central limit theorem for @math that is based on a new variant of Stein's method.
  @cite_2: 
  @cite_5: Binary matrices, and their associated submatrices of 1s, play a central role in the study of random bipartite graphs and in core data mining problems such as frequent itemset mining (FIM). Motivated by these connections, this paper addresses several statistical questions regarding submatrices of 1s in a random binary matrix with independent Bernoulli entries. We establish a three-point concentration result, and a related probability bound, for the size of the largest square submatrix of 1s in a square Bernoulli matrix, and extend these results to non-square matrices and submatrices with fixed aspect ratios. We then consider the noise sensitivity of frequent itemset mining under a simple binary additive noise model, and show that, even at small noise levels, large blocks of 1s leave behind fragments of only logarithmic size. As a result, standard FIM algorithms, which search only for submatrices of 1s, cannot directly recover such blocks when noise is present. On the positive side, we show that an error-tolerant frequent itemset criterion can recover a submatrix of 1s against a background of 0s plus noise, even when the size of the submatrix of 1s is very small.
  @cite_31: We consider the problem of detecting whether or not in a given sensor network, there is a cluster of sensors which exhibit an "unusual behavior." Formally, suppose we are given a set of nodes and attach a random variable to each node. We observe a realization of this process and want to decide between the following two hypotheses: under the null, the variables are i.i.d. standard normal; under the alternative, there is a cluster of variables that are i.i.d. normal with positive mean and unit variance, while the rest are i.i.d. standard normal. We also address surveillance settings where each sensor in the network collects information over time. The resulting model is similar, now with a time series attached to each node. We again observe the process over time and want to decide between the null, where all the variables are i.i.d. standard normal; and the alternative, where there is an emerging cluster of i.i.d. normal variables with positive mean and unit variance. The growth models used to represent the emerging cluster are quite general, and in particular include cellular automata used in modelling epidemics. In both settings, we consider classes of clusters that are quite general, for which we obtain a lower bound on their respective minimax detection rate, and show that some form of scan statistic, by far the most popular method in practice, achieves that same rate within a logarithmic factor. Our results are not limited to the normal location model, but generalize to any one-parameter exponential family when the anoma- lous clusters are large enough.
summary: A rich line of research in statistics addresses the problem of identifying the non-zero entries in a sparse vector (or matrix) @math from observations @math where @math has typically i.i.d. standard Gaussian entries. In particular @cite_38 @cite_2 @cite_31 @cite_0 study cases in which the sparsity pattern of @math is structured'. For instance, we can take @math a matrix with @math if @math and @math otherwise. This fits the framework studied in this paper, for @math the complete graph and @math , @math . This literature however disregards computational considerations. Greedy search methods were developed in several papers, see e.g. @cite_5 @cite_22 .