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In the field of statistical learning and data analysis, estimating precision matrices (i.e., the inverse of covariance matrices) is a critical task, particularly for understanding dependency structures among variables. However, traditional…
We propose a penalized likelihood framework for estimating multiple precision matrices from different classes. Most existing methods either incorporate no information on relationships between the precision matrices, or require this…
We consider the matrix completion problem where the aim is to esti-mate a large data matrix for which only a relatively small random subset of its entries is observed. Quite popular approaches to matrix completion problem are iterative…
The computation of the sparse principal component of a matrix is equivalent to the identification of its principal submatrix with the largest maximum eigenvalue. Finding this optimal submatrix is what renders the problem…
Estimating covariance matrices is a problem of fundamental importance in multivariate statistics. In practice it is increasingly frequent to work with data matrices $X$ of dimension $n\times p$, where $p$ and $n$ are both large. Results…
In this article, we show multiple inequalities for the singular values of the difference of matrix means. The obtained results refine and complement some well established results in the literature. Although we target singular values…
Given a large data matrix, sparsifying, quantizing, and/or performing other entry-wise nonlinear operations can have numerous benefits, ranging from speeding up iterative algorithms for core numerical linear algebra problems to providing…
Efficiently computing a subset of a correlation matrix consisting of values above a specified threshold is important to many practical applications. Real-world problems in genomics, machine learning, finance other applications can produce…
We determine the rank of a random matrix over an arbitrary field with prescribed numbers of non-zero entries in each row and column. As an application we obtain a formula for the rate of low-density parity check codes. This formula…
In the construction of low-rank matrix approximation and maximum element search it is effective to use maxvol algorithm. Nevertheless, even in the case of rank 1 approximation the algorithm does not always converge to the maximum matrix…
In this paper, we provide an approach to clustering relational matrices whose entries correspond to either similarities or dissimilarities between objects. Our approach is based on the value of information, a parameterized,…
Block sparsity is an important parameter in many algorithms to successfully recover block sparse signals under the framework of compressive sensing. However, it is often unknown and needs to be estimated. Recently there emerges a few…
Efficient solutions of large-scale, ill-conditioned and indefinite algebraic equations are ubiquitously needed in numerous computational fields, including multiphysics simulations, machine learning, and data science. Because of their…
The matrices used in many computational settings are naturally sparse, holding a small percentage of nonzero elements. Storing such matrices in specialized sparse formats enables algorithms that avoid wasting computation on zeros,…
Randomized algorithms provide solutions to two ubiquitous problems: (1) the distributed calculation of a principal component analysis or singular value decomposition of a highly rectangular matrix, and (2) the distributed calculation of a…
Singular value decompositions of matrices are widely used in numerical linear algebra with many applications. In this paper, we extend the notion of singular value decompositions to finite complexes of real vector spaces. We provide two…
Motivated by a neuroscience application we study the problem of statistical estimation of a high-dimensional covariance matrix with a block structure. The block model embeds a structural assumption: the population of items (neurons) can be…
Linear algebraic expressions are the essence of many computationally intensive problems, including scientific simulations and machine learning applications. However, translating high-level formulations of these expressions to efficient…
Matrix completion constantly receives tremendous attention from many research fields. It is commonly applied for recommender systems such as movie ratings, computer vision such as image reconstruction or completion, multi-task learning such…
Sparse principal component analysis addresses the problem of finding a linear combination of the variables in a given data set with a sparse coefficients vector that maximizes the variability of the data. This model enhances the ability to…