Related papers: Multiple typical ranks in matrix completion

After the phenomenal success of the PageRank algorithm, many researchers have extended the PageRank approach to ranking graphs with richer structures beside the simple linkage structure. In some scenarios we have to deal with…

Let G = (A U P, E) be a bipartite graph where A denotes a set of agents, P denotes a set of posts and ranks on the edges denote preferences of the agents over posts. A matching M in G is rank-maximal if it matches the maximum number of…

Matrix completion tackles the task of predicting missing values in a low-rank matrix based on a sparse set of observed entries. It is often assumed that the observation pattern is generated uniformly at random or has a very specific…

Mathematically characterizing the implicit regularization induced by gradient-based optimization is a longstanding pursuit in the theory of deep learning. A widespread hope is that a characterization based on minimization of norms may…

Matrix factorization models have been extensively studied as a valuable test-bed for understanding the implicit biases of overparameterized models. Although both low nuclear norm and low rank regularization have been studied for these…

Conventional ways to solve optimization problems on low-rank matrix sets which appear in great number of applications ignore its underlying structure of an algebraic variety and existence of singular points. This leads to appearance of…

We consider the symmetric Toeplitz matrix completion problem, whose matrix under consideration possesses specific row and column structures. This problem, which has wide application in diverse areas, is well-known to be computationally…

In this paper we study the joint/generalized spectral radius of a finite set of matrices in terms of its rank-one approximation by singular value decomposition. In the first part of the paper, we show that any finite set of matrices with at…

An n\times n matrix M is called a fooling-set matrix of size n, if its diagonal entries are nonzero, whereas for every k\ne \ell we have M_{k,\ell} M_{\ell,k} = 0. Dietzfelbinger, Hromkovi\v{c}, and Schnitger (1996) showed that n \le (\rk…

The process of rank aggregation is intimately intertwined with the structure of skew-symmetric matrices. We apply recent advances in the theory and algorithms of matrix completion to skew-symmetric matrices. This combination of ideas…

This paper introduces a novel algorithm to approximate the matrix with minimum nuclear norm among all matrices obeying a set of convex constraints. This problem may be understood as the convex relaxation of a rank minimization problem, and…

Saturating sets are combinatorial objects in projective spaces over finite fields that have been intensively investigated in the last three decades. They are related to the so-called covering problem of codes in the Hamming metric. In this…

Consider a movie recommendation system where apart from the ratings information, side information such as user's age or movie's genre is also available. Unlike standard matrix completion, in this setting one should be able to predict…

In this paper we give an explicit solution to the rank constrained matrix approximation in Frobenius norm, which is a generalization of the classical approximation of an m by n matrix A by a matrix of rank k at most.

The paper explores the concept of the rank of a bicomplex matrix, delving into four distinct types of ranks and investigating conditions under which these ranks are equivalent. It also defines and analyzes the concept of idempotent row…

We propose a computational framework for computing low-rank approximations to the ensemble of solutions of a parametrized system of the form $A(\xi)x(\xi)+g(x(\xi))=b(\xi)$ for multiple parameter values. The central idea is to reinterpret…

Robust Principal Component Analysis (PCA) (Candes et al., 2011) and low-rank matrix completion (Recht et al., 2010) are extensions of PCA to allow for outliers and missing entries respectively. It is well-known that solving these problems…

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…

Low-rank matrix completion has been studied extensively under various type of categories. The problem could be categorized as noisy completion or exact completion, also active or passive completion algorithms. In this paper we focus on…

We study the completion of approximately low rank matrices with entries missing not at random (MNAR). In the context of typical large-dimensional statistical settings, we establish a framework for the performance analysis of the nuclear…