相关论文: On the compressibility of tensors

We describe a simple, black-box compression format for tensors with a multiscale structure. By representing the tensor as a sum of compressed tensors defined on increasingly coarse grids, we capture low-rank structures on each grid-scale,…

We provide new upper and lower bounds on the minimum possible ratio of the spectral and Frobenius norms of a (partially) symmetric tensor. In the particular case of general tensors our result recovers a known upper bound. For symmetric…

In this paper we propose new techniques to sample arbitrary third-order tensors, with an objective of speeding up tensor algorithms that have recently gained popularity in machine learning. Our main contribution is a new way to select, in a…

We consider 3D free-boundary compressible elastodynamic system under the Rayleigh-Taylor sign condition. It describes the motion of an isentropic inviscid elastic medium with moving boundary. The deformation tensor satisfies the neo-Hookean…

Tensor decomposition on big data has attracted significant attention recently. Among the most popular methods is a class of algorithms that leverages compression in order to reduce the size of the tensor and potentially parallelize…

Compressed sensing extends from the recovery of sparse vectors from undersampled measurements via efficient algorithms to the recovery of matrices of low rank from incomplete information. Here we consider a further extension to the…

Third-order tensors are widely used as a mathematical tool for modeling physical properties of media in solid state physics. In most cases, they arise as constitutive tensors of proportionality between basic physics quantities. The…

An anisotropic elasticity tensor can be approximated by the closest tensor belonging to a higher symmetry class. The closeness of tensors depends on the choice of a criterion. We compare the closest isotropic tensors obtained using four…

This chapter studies the problem of decomposing a tensor into a sum of constituent rank one tensors. While tensor decompositions are very useful in designing learning algorithms and data analysis, they are NP-hard in the worst-case. We will…

Our interest lies in the recoverability properties of compressed tensors under the \textit{canonical polyadic decomposition} (CPD) model. The considered problem is well-motivated in many applications, e.g., hyperspectral image and video…

We present a method for converting tensors into tensor train format based on actions of the tensor as a vector-valued multilinear function. Existing methods for constructing tensor trains require access to "array entries" of the tensor and…

We discuss how recently discovered techniques and tools from compressed sensing can be used in tensor decompositions, with a view towards modeling signals from multiple arrays of multiple sensors. We show that with appropriate bounds on a…

Electron repulsion integral tensor has ubiquitous applications in quantum chemistry calculations. In this work, we propose an algorithm which compresses the electron repulsion tensor into the tensor hypercontraction format with…

The closest tensors of higher symmetry classes are derived in explicit form for a given elasticity tensor of arbitrary symmetry. The mathematical problem is to minimize the elastic length or distance between the given tensor and the closest…

This paper presents a method to analyze the powers of a given trilinear form (a special kind of algebraic constructions also called a tensor) and obtain upper bounds on the asymptotic complexity of matrix multiplication. Compared with…

Deep neural networks generalize well on unseen data though the number of parameters often far exceeds the number of training examples. Recently proposed complexity measures have provided insights to understanding the generalizability in…

Coupled tensor decomposition reveals the joint data structure by incorporating priori knowledge that come from the latent coupled factors. The tensor ring (TR) decomposition is invariant under the permutation of tensors with different mode…

Tensor programs often need to process large tensors (vectors, matrices, or higher order tensors) that require a specialized storage format for their memory layout. Several such layouts have been proposed in the literature, such as the…

Many real-world datasets are represented as tensors, i.e., multi-dimensional arrays of numerical values. Storing them without compression often requires substantial space, which grows exponentially with the order. While many tensor…

We give constructions of n^k x n^k x n tensors of rank at least 2n^k - O(n^(k-1)). As a corollary we obtain an [n]^r shaped tensor with rank at least 2n^(r/2) - O(n^(r/2)-1) when r is odd. The tensors are constructed from a simple recursive…