Related papers: Sparse Principal Component of a Rank-deficient Mat…

Given an n x n matrix A, we present a simple, element-wise sparsification algorithm that zeroes out all sufficiently small elements of A and then retains some of the remaining elements with probabilities proportional to the square of their…

We consider the problem of estimating a rank-one nonsymmetric matrix under additive white Gaussian noise. The matrix to estimate can be written as the outer product of two vectors and we look at the special case in which both vectors are…

The problem of identifying sparse solutions for the link structure and dynamics of an unknown linear, time-invariant network is posed as finding sparse solutions x to Ax=b. If the sensing matrix A satisfies a rank condition, this problem…

Variable selection for sparse linear regression is the problem of finding, given an m x p matrix B and a target vector y, a sparse vector x such that Bx approximately equals y. Assuming a standard complexity hypothesis, we show that no…

Generalized inverses play a fundamental role in numerical linear algebra, particularly when matrices are rectangular, singular, or rank deficient. Even when the input matrix is sparse, generalized inverses such as the M-P pseudoinverse are…

Here, we give an algorithm for deciding if the nonnegative rank of a matrix $M$ of dimension $m \times n$ is at most $r$ which runs in time $(nm)^{O(r^2)}$. This is the first exact algorithm that runs in time singly-exponential in $r$. This…

The coresets approach, also called subsampling or subset selection, aims to select a subsample as a surrogate for the observed sample and has found extensive applications in large-scale data analysis. Existing coresets methods construct the…

Sparse Principal Component Analysis (sparse PCA) is a fundamental dimension-reduction tool that enhances interpretability in various high-dimensional settings. An important variant of sparse PCA studies the scenario when samples are…

Previous versions of sparse principal component analysis (PCA) have presumed that the eigen-basis (a $p \times k$ matrix) is approximately sparse. We propose a method that presumes the $p \times k$ matrix becomes approximately sparse after…

The sparse generalized eigenvalue problem arises in a number of standard and modern statistical learning models, including sparse principal component analysis, sparse Fisher discriminant analysis, and sparse canonical correlation analysis.…

Given a matrix A \in R^{m x n}, we present a randomized algorithm that sparsifies A by retaining some of its elements by sampling them according to a distribution that depends on both the square and the absolute value of the entries. We…

A cofactor representation of an ideal element, that is, a representation in terms of the generators, can be considered as a certificate for ideal membership. Such a representation is typically not unique, and some can be a lot more…

We consider the inverse problem of finding a magnitude-symmetric matrix (matrix with opposing off-diagonal entries equal in magnitude) with a prescribed set of principal minors. This problem is closely related to the theory of recognizing…

Sparse principal component analysis (PCA) is an important technique for dimensionality reduction of high-dimensional data. However, most existing sparse PCA algorithms are based on non-convex optimization, which provide little guarantee on…

We analyze a practical algorithm for sparse PCA on incomplete and noisy data under a general non-random sampling scheme. The algorithm is based on a semidefinite relaxation of the $\ell_1$-regularized PCA problem. We provide theoretical…

We propose a new variant of nonnegative matrix factorization (NMF), combining separability and sparsity assumptions. Separability requires that the columns of the first NMF factor are equal to columns of the input matrix, while sparsity…

Ranking items based on pairwise comparisons is common, from using match outcomes to rank sports teams to using purchase or survey data to rank consumer products. Statistical inference-based methods such as the Bradley-Terry model, which…

Let $\mathbf{A}_{n,m;k}$ be a random $n \times m$ matrix with entries from some field $\mathbb{F}$ where there are exactly $k$ non-zero entries in each column, whose locations are chosen independently and uniformly at random from the set of…

In this paper, we first introduce the concept of Laurent differentially essential systems and give a criterion for Laurent differentially essential systems in terms of their supports. Then the sparse differential resultant for a Laurent…

We study sparse principal component analysis in the high-dimensional, sample-limited regime, aiming to recover a leading component supported on a few coordinates. Despite extensive progress, most methods and analyses are tailored to the…