Related papers: Approximate Matrix Multiplication with Application…
The task of reconstructing a matrix given a sample of observedentries is known as the matrix completion problem. It arises ina wide range of problems, including recommender systems, collaborativefiltering, dimensionality reduction, image…
Minimizing the rank of a matrix subject to constraints is a challenging problem that arises in many applications in control theory, machine learning, and discrete geometry. This class of optimization problems, known as rank minimization, is…
The problem of low rank approximation is ubiquitous in science. Traditionally this problem is solved in unitary invariant norms such as Frobenius or spectral norm due to existence of efficient methods for building approximations. However,…
Matrix completion is the problem of recovering a low rank matrix by observing a small fraction of its entries. A series of recent works [KOM12,JNS13,HW14] have proposed fast non-convex optimization based iterative algorithms to solve this…
We introduce a "learning-based" algorithm for the low-rank decomposition problem: given an $n \times d$ matrix $A$, and a parameter $k$, compute a rank-$k$ matrix $A'$ that minimizes the approximation loss $\|A-A'\|_F$. The algorithm uses a…
Low-rank matrix regression is a fundamental problem in data science with various applications in systems and control. Nuclear norm regularization has been widely applied to solve this problem due to its convexity. However, it suffers from…
Nonlinear dimensionality reduction or, equivalently, the approximation of high-dimensional data using a low-dimensional nonlinear manifold is an active area of research. In this paper, we will present a thematically different approach to…
Low-rank matrices are pervasive throughout statistics, machine learning, signal processing, optimization, and applied mathematics. In this paper, we propose a novel and user-friendly Euclidean representation framework for low-rank matrices.…
This paper studies the matrix completion problem under arbitrary sampling schemes. We propose a new estimator incorporating both max-norm and nuclear-norm regularization, based on which we can conduct efficient low-rank matrix recovery…
Whether the goal is to analyze voting behavior, locate facilities, or recommend products, the problem of translating between (ordinal) rankings and (numerical) utilities arises naturally in many contexts. This task is commonly approached by…
The problem of recovering the configuration of points from their partial pairwise distances, referred to as the Euclidean Distance Matrix Completion (EDMC) problem, arises in a broad range of applications, including sensor network…
We present an analysis of sets of matrices with rank less than or equal to a specified number $s$. We provide a simple formula for the normal cone to such sets, and use this to show that these sets are prox-regular at all points with rank…
In this paper, we address the problem of estimating transport surplus (a.k.a. matching affinity) in high dimensional optimal transport problems. Classical optimal transport theory specifies the matching affinity and determines the optimal…
Work on approximate linear algebra has led to efficient distributed and streaming algorithms for problems such as approximate matrix multiplication, low rank approximation, and regression, primarily for the Euclidean norm $\ell_2$. We study…
The approximate nearest neighbor problem ($\epsilon$-ANN) in high dimensional Euclidean space has been mainly addressed by Locality Sensitive Hashing (LSH), which has polynomial dependence in the dimension, sublinear query time, but…
We study random submatrices of a large matrix A. We show how to approximately compute A from its random submatrix of the smallest possible size O(r log r) with a small error in the spectral norm, where r = ||A||_F^2 / ||A||_2^2 is the…
We study the convergence of specific inexact alternating projections for two non-convex sets in a Euclidean space. The $\sigma$-quasioptimal metric projection ($\sigma \geq 1$) of a point $x$ onto a set $A$ consists of points in $A$ the…
We study the problem of supervised learning a metric space under discriminative constraints. Given a universe $X$ and sets ${\cal S}, {\cal D}\subset {X \choose 2}$ of similar and dissimilar pairs, we seek to find a mapping $f:X\to Y$, into…
Structured low-rank approximation is the problem of minimizing a weighted Frobenius distance to a given matrix among all matrices of fixed rank in a linear space of matrices. We study exact solutions to this problem by way of computational…
We propose a new technique for constructing low-rank approximations of matrices that arise in kernel methods for machine learning. Our approach pairs a novel automatically constructed analytic expansion of the underlying kernel function…