Related papers: The Nondecreasing Rank
We consider the problem of noisy matrix completion, in which the goal is to reconstruct a structured matrix whose entries are partially observed in noise. Standard approaches to this underdetermined inverse problem are based on assuming…
This paper studies the issues about tensors. Three typical kinds of tensor decomposition are mentioned. Among these decompositions, the t-SVD is proposed in this decade. Different definitions of rank derive from tensor decompositions. Based…
A tensor defined over a finite field $\mathbb{F}$ has low analytic rank if the distribution of its values differs significantly from the uniform distribution. An order $d$ tensor has partition rank 1 if it can be written as a product of two…
This paper proposes a novel formulation of the tensor completion problem to impute missing entries of data represented by tensors. The formulation is introduced in terms of tensor train (TT) rank which can effectively capture global…
Finding the rank of a tensor is a problem that has many applications. Unfortunately it is often very difficult to determine the rank of a given tensor. Inspired by the heuristics of convex relaxation, we consider the nuclear norm instead of…
Non-Negative Matrix Factorization (NMF) is a widely used dimension reduction method that factorizes a non-negative data matrix into two lower dimensional non-negative matrices: One is the basis or feature matrix which consists of the…
Matrix completion algorithms recover a low rank matrix from a small fraction of the entries, each entry contaminated with additive errors. In practice, the singular vectors and singular values of the low rank matrix play a pivotal role for…
Low rank matrix and tensor completion problems are to recover the incomplete two and higher order data by using their low rank structures. The essential problem in the matrix and tensor completion problems is how to improve the efficiency.…
Tensors are often studied by introducing preorders such as restriction and degeneration: the former describes transformations of the tensors by local linear maps on its tensor factors; the latter describes transformations where the local…
Compressing neural networks is a key step when deploying models for real-time or embedded applications. Factorizing the model's matrices using low-rank approximations is a promising method for achieving compression. While it is possible to…
We prove (without exceptions) the existence of irredundant tensor decompositions with the number of addenda equal to rank $+1$. We also discuss the existence of decompositions with more than the tensor rank terms, which are concise, while…
Given an input matrix polynomial whose coefficients are floating point numbers, we consider the problem of finding the nearest matrix polynomial which has rank at most a specified value. This generalizes the problem of finding a nearest…
Tensor rank and low-rank tensor decompositions have many applications in learning and complexity theory. Most known algorithms use unfoldings of tensors and can only handle rank up to $n^{\lfloor p/2 \rfloor}$ for a $p$-th order tensor in…
The subrank of tensors is a measure of how much a tensor can be ''diagonalized''. This parameter was introduced by Strassen to study fast matrix multiplication algorithms in algebraic complexity theory and is closely related to many central…
We consider the problem of low-rank decomposition of incomplete multiway tensors. Since many real-world data lie on an intrinsically low dimensional subspace, tensor low-rank decomposition with missing entries has applications in many data…
The Nystr\"om method offers an effective way to obtain low-rank approximation of SPD matrices, and has been recently extended and analyzed to nonsymmetric matrices (leading to the generalized Nystr\"om method). It is a randomized,…
Factor model is an appealing and effective analytic tool for high-dimensional time series, with a wide range of applications in economics, finance and statistics. This paper develops two criteria for the determination of the number of…
It is well-known that tensor decompositions show separations, that is, that constraints on local terms (such as positivity) may entail an arbitrarily high cost in their representation. Here we show that many of these separations disappear…
In this paper we study various versions of extension complexity for polygons through the study of factorization ranks of their slack matrices. In particular, we develop a new asymptotic lower bound for their nonnegative rank, shortening the…
Nonnegative tensor factorization has applications in statistics, computer vision, exploratory multiway data analysis and blind source separation. A symmetric nonnegative tensor, which has a symmetric nonnegative factorization, is called a…