Rank constrained matrix best approximation problem

Generalized Rank-Constrained Matrix Approximations

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× n matrix A by a matrix of rank k at most. 2000 Mathematics Subject Classification: 15A18.

Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem

There has been continued interest in seeking a theorem describing optimal low-rank approximations to tensors of order 3 or higher, that parallels the Eckart–Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, tensors of order 3 or higher can fail to have best rank-r approximations. The phenomenon is much more...

Local Low-Rank Matrix Approximation

Matrix approximation is a common tool in recommendation systems, text mining, and computer vision. A prevalent assumption in constructing matrix approximations is that the partially observed matrix is of low-rank. We propose a new matrix approximation model where we assume instead that the matrix is locally of low-rank, leading to a representation of the observed matrix as a weighted sum of low...

A Matrix Rank Problem

On best rank one approximation of tensors

Today, compact and reduced data representations using low rank data approximation are common to represent high-dimensional data sets in many application areas as for example genomics, multimedia, quantum chemistry, social networks or visualization. In order to produce such low rank data representations, the input data is typically approximated by so-called alternating least squares (ALS) algori...