Related papers: Energy Landscape for large average submatrix detec…
In this paper, we study the problem of detecting multiple hidden submatrices in a large Gaussian random matrix when the planted signal is inhomogeneous across entries. Under the null hypothesis, the observed matrix has independent and…
We consider the high-dimensional inference problem where the signal is a low-rank symmetric matrix which is corrupted by an additive Gaussian noise. Given a probabilistic model for the low-rank matrix, we compute the limit in the large…
We consider the following general hidden hubs model: an $n \times n$ random matrix $A$ with a subset $S$ of $k$ special rows (hubs): entries in rows outside $S$ are generated from the probability distribution $p_0 \sim N(0,\sigma_0^2)$; for…
We compute exact asymptotic results for the probability of the occurrence of large deviations of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary and symplectic ensembles. In particular, we…
In this paper we present a method for learning the parameters of a mixture of $k$ identical spherical Gaussians in $n$-dimensional space with an arbitrarily small separation between the components. Our algorithm is polynomial in all…
We consider the problem of detecting the presence of a submatrix with larger-than-usual values in a large data matrix. This problem was considered in (Butucea and Ingster, 2013) under a one-parameter exponential family, and one of the test…
Positive semi-definite matrices commonly occur as normal matrices of least squares problems in statistics or as kernel matrices in machine learning and approximation theory. They are typically large and dense. Thus algorithms to solve…
We consider general Gaussian latent tree models in which the observed variables are not restricted to be leaves of the tree. Extending related recent work, we give a full semi-algebraic description of the set of covariance matrices of any…
A local search algorithm operating on an instance of a Boolean constraint satisfaction problem (in particular, k-SAT) can be viewed as a stochastic process traversing successive adjacent states in an ``energy landscape'' defined by the…
We consider a matrix-valued Gaussian sequence model, that is, we observe a sequence of high-dimensional $M \times N$ matrices of heterogeneous Gaussian random variables $x_{ij,k}$ for $i \in\{1,...,M\}$, $j \in \{1,...,N\}$ and $k \in…
Motivated by current interest in understanding statistical properties of random landscapes in high-dimensional spaces, we consider a model of the landscape in $\mathbb{R}^N$ obtained by superimposing $M>N$ plane waves of random wavevectors…
The classical Gaussian ensembles of random matrices can be constructed by maximizing Boltzmann-Gibbs-Shannon's entropy, S_{BGS} = - \int d{\bf H} [P({\bf H})] \ln [P({\bf H})], with suitable constraints. Here we construct and analyze…
We study subset selection for matrices defined as follows: given a matrix $\matX \in \R^{n \times m}$ ($m > n$) and an oversampling parameter $k$ ($n \le k \le m$), select a subset of $k$ columns from $\matX$ such that the pseudo-inverse of…
Gaussian elimination is the most popular technique for solving a dense linear system. Large errors in this procedure can occur in floating point arithmetic when the matrix's growth factor is large. In the study of numerical linear algebra,…
We study the largest eigenvalue of a Gaussian random symmetric matrix $X_n$, with zero-mean, unit variance entries satisfying the condition $\sup_{(i, j) \ne (i', j')}|\mathbb{E}[X_{ij} X_{i'j'}]| = O(n^{-(1 + \varepsilon)})$, where…
Generalized linear models (GLMs) arise in high-dimensional machine learning, statistics, communications and signal processing. In this paper we analyze GLMs when the data matrix is random, as relevant in problems such as compressed sensing,…
There is growing interest in improving our algorithmic understanding of fundamental statistical problems such as mean estimation, driven by the goal of understanding the limits of what we can extract from valuable data. The state of the art…
A common task in physics, information theory, and other fields is the analysis of properties of subsystems of a given system. Given the covariance matrix $M$ of a system of $n$ coupled variables, the covariance matrices of the subsystems…
The problem of non-monotone $k$-submodular maximization under a knapsack constraint ($\kSMK$) over the ground set size $n$ has been raised in many applications in machine learning, such as data summarization, information propagation, etc.…
We consider the double scaling limit in the random matrix ensemble with an external source $\frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM$ defined on $n\times n$ Hermitian matrices, where $A$ is a diagonal matrix with two eigenvalues $\pm a$ of…