Low-rank Tensor Standard Forms and Limits∗
نویسنده
چکیده
Abstract. Despite the growing use of tensor-based methods, tensors remain mysterious and misunderstood objects. In this work we develop a standard form for low-rank tensors that allows one to focus on the important defining parameters (internal angles and comparative sizes) while ignoring unimportant parameters (separable unitary transformations). This standard form eases the analysis of tensor approximation algorithms and allows various examples from the literature to be compared. We then use this standard form to characterize limits of rank-2 and rank-3 tensors. As side results, we prove that the minimal rank of a Laplacian-like tensor in d dimensions is d and that the minimal rank over the reals of the sine of the sum of d variables is d.
منابع مشابه
On the Tensor Svd and Optimal Low Rank Orthogonal Approximations of Tensors
Abstract. It is known that a high order tensor does not necessarily have an optimal low rank approximation, and that a tensor might not be orthogonally decomposable (i.e., admit a tensor SVD). We provide several sufficient conditions which lead to the failure of the tensor SVD, and characterize the existence of the tensor SVD with respect to the Higher Order SVD (HOSVD) of a tensor. In face of ...
متن کاملA General Model for Robust Tensor Factorization with Unknown Noise
Because of the limitations of matrix factorization, such as losing spatial structure information, the concept of low-rank tensor factorization (LRTF) has been applied for the recovery of a low dimensional subspace from high dimensional visual data. The low-rank tensor recovery is generally achieved by minimizing the loss function between the observed data and the factorization representation. T...
متن کاملLow-Rank Approximation and Completion of Positive Tensors
Unlike the matrix case, computing low-rank approximations of tensors is NP-hard and numerically ill-posed in general. Even the best rank-1 approximation of a tensor is NP-hard. In this paper, we use convex optimization to develop polynomial-time algorithms for low-rank approximation and completion of positive tensors. Our approach is to use algebraic topology to define a new (numerically well-p...
متن کاملDiscretized Dynamical Low-Rank Approximation in the Presence of Small Singular Values
Low-rank approximations to large time-dependent matrices and tensors are the subject of this paper. These matrices and tensors are either given explicitly or are the unknown solutions of matrix and tensor differential equations. Based on splitting the orthogonal projection onto the tangent space of the low-rank manifold, novel time integrators for obtaining approximations by low-rank matrices a...
متن کاملDiscretized Dynamical Low-rank Approximation in the Presence of Small Singular
Low-rank approximations to large time-dependent matrices and tensors are the subject of this paper. These matrices and tensors either are given explicitly or are the unknown solutions of matrix and tensor differential equations. Based on splitting the orthogonal projection onto the tangent space of the low-rank manifold, novel time integrators for obtaining approximations by low-rank matrices a...
متن کامل