Related papers: Phase transitions in spiked matrix estimation: inf…
The spiked model is an important special case of the Wishart ensemble, and a natural generalization of the white Wishart ensemble. Mathematically, it can be defined on three kinds of variables: the real, the complex and the quaternion. For…
We consider the problem of estimating a rank-one matrix in Gaussian noise under a probabilistic model for the left and right factors of the matrix. The probabilistic model can impose constraints on the factors including sparsity and…
This work concerns elementwise-transformations of spiked matrices: $Y_n = n^{-1/2} f( \sqrt{n} X_n + Z_n)$. Here, $f$ is a function applied elementwise, $X_n$ is a low-rank signal matrix, and $Z_n$ is white noise. We find that principal…
We consider a sparse linear regression model Y=X\beta^{*}+W where X has a Gaussian entries, W is the noise vector with mean zero Gaussian entries, and \beta^{*} is a binary vector with support size (sparsity) k. Using a novel conditional…
These lecture notes provide a comprehensive, self-contained introduction to the analysis of Wishart matrix moments. This study may act as an introduction to some particular aspects of random matrix theory, or as a self-contained exposition…
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…
Many problems in statistics and machine learning require the reconstruction of a rank-one signal matrix from noisy data. Enforcing additional prior information on the rank-one component is often key to guaranteeing good recovery…
We study low-rank matrix estimation for a generic inhomogeneous output channel through which the matrix is observed. This generalizes the commonly considered spiked matrix model with homogeneous noise to include for instance the dense…
We consider high dimensional Wishart matrices $\mathbb{X} \mathbb{X}^{\top}$ where the entries of $\mathbb{X} \in {\mathbb{R}^{n \times d}}$ are i.i.d. from a log-concave distribution. We prove an information theoretic phase transition:…
Consider the problem of estimating a low-rank matrix when its entries are perturbed by Gaussian noise. If the empirical distribution of the entries of the spikes is known, optimal estimators that exploit this knowledge can substantially…
We consider the high-dimensional inference problem where the signal is a low-rank 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 dimension…
We analyse the matrix factorization problem. Given a noisy measurement of a product of two matrices, the problem is to estimate back the original matrices. It arises in many applications such as dictionary learning, blind matrix…
We study optimal estimation for sparse principal component analysis when the number of non-zero elements is small but on the same order as the dimension of the data. We employ approximate message passing (AMP) algorithm and its state…
This article is an extended version of previous work of the authors [40, 41] on low-rank matrix estimation in the presence of constraints on the factors into which the matrix is factorized. Low-rank matrix factorization is one of the basic…
The spiked Wigner ensemble is a prototypical model for high-dimensional inference. We study the spectral properties of an inhomogeneous rank-one spiked Wigner model in which the variance of each entry of the noise matrix is itself a random…
Despite the widespread use and success of machine-learning techniques for detecting phase transitions from data, their working principle and fundamental limits remain elusive. Here, we explain the inner workings and identify potential…
We study the phase transition phenomenon inherent in the shuffled (permuted) regression problem, which has found numerous applications in databases, privacy, data analysis, etc. In this study, we aim to precisely identify the locations of…
Structural matrix-variate observations routinely arise in diverse fields such as multi-layer network analysis and brain image clustering. While data of this type have been extensively investigated with fruitful outcomes being delivered, the…
We study the computational phase transition in a multi-frequency group synchronization problem, where pairwise relative measurements of group elements are observed across multiple frequency channels and corrupted by Gaussian noise. Using…
Many data-analysis problems involve large dense matrices that describe the covariance of stationary noise processes; the computational cost of inverting these matrices, or equivalently of solving linear systems that contain them, is often a…