Related papers: On Testing Kronecker Product Structure in Tensor F…
Decompositions of tensors into factor matrices, which interact through a core tensor, have found numerous applications in signal processing and machine learning. A more general tensor model which represents data as an ordered network of…
It is known that the Kronecker coefficient of three partitions is a bounded and weakly increasing sequence if one increases the first part of all three partitions. Furthermore if the first parts of partitions \lambda,\mu are big enough then…
We study the computational model where we can access a matrix $\mathbf{A}$ only by computing matrix-vector products $\mathbf{A}\mathrm{x}$ for vectors of the form $\mathrm{x} = \mathrm{x}_1 \otimes \cdots \otimes \mathrm{x}_q$. We prove…
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…
Tensor decompositions have become a central tool in data science, with applications in areas such as data analysis, signal processing, and machine learning. A key property of many tensor decompositions, such as the canonical polyadic…
The proposed article aims at offering a comprehensive tutorial for the computational aspects of structured matrix and tensor factorization. Unlike existing tutorials that mainly focus on {\it algorithmic procedures} for a small set of…
In this text we review a few structural properties of matrix models that should at least partly generalize to random tensor models. We review some aspects of the loop equations for matrix models and their algebraic counterpart for tensor…
We study two new families of symmetric functions arising from a species-theoretic construction motivated by cycle structure. For each partition of $n$, we define two combinatorial species that decompose into molecules indexed by the same…
An observed $K$-dimensional series $\left\{ y_{n}\right\} _{n=1}^{N}$ is expressed in terms of a lower $p$-dimensional latent series called factors $f_{n}$ and random noise $\varepsilon_{n}$. The equation, $y_{n}=Qf_{n}+\varepsilon_{n}$ is…
A novel tensor decomposition framework, termed Tensor Star (TS) decomposition, is proposed which represents a new type of tensor network decomposition based on tensor contractions. This is achieved by connecting the core tensors in a ring…
Using the block vec matrix, I give a necessary and sufficient condition for factorization of a matrix into the Kronecker product of two other matrices. As a consequence, I obtain an elementary algorithmic procedure to decide whether a…
We derive a Kronecker product approximation for the micromagnetic long range interactions in a collocation framework by means of separable sinc quadrature. Evaluation of this operator for structured tensors (Canonical format, Tucker format,…
We review the properties of the Kronecker (direct, or tensor) product of square matrices $A \otimes B \otimes C \cdots$ in terms of Hubbard operators. In its simplest form, a Hubbard operator $X_n^{i,j}$ can be expressed as the $n$-square…
We introduce and extend the outer product and contractive product of tensors and matrices, and present some identities in terms of these products. We offer tensor expressions of derivatives of tensors, focus on the tensor forms of…
We prove a generalization to Jennrich's uniqueness theorem for tensor decompositions in the undercomplete setting. Our uniqueness theorem is based on an alternative definition of the standard tensor decomposition, which we call…
A Kronecker coefficient is the multiplicity of an irreducible representation of a finite group $G$ in a tensor product of irreducible representations. We define Kronecker Hecke algebras and use them as a tool to study Kronecker coefficients…
Given a vector-space $~V~$ which is the tensor product of vector-spaces $A$ and $B$, we reconstruct $A$ and $B$ from the family of simple tensors $a{\otimes}b$ within $V$. In an application to quantum mechanics, one would be reconstructing…
Decomposing tensors into orthogonal factors is a well-known task in statistics, machine learning, and signal processing. We study orthogonal outer product decompositions where the factors in the summands in the decomposition are required to…
We discuss extended definitions of linear and multilinear operations such as Kronecker, Hadamard, and contracted products, and establish links between them for tensor calculus. Then we introduce effective low-rank tensor approximation…
The annulus comes with a "stacking" operation which glues two annuli into one. This provides a tensor product structure on the category of boundary values $Z_{\text{CY}}(\text{Ann})$ associated to the annulus in an extended Crane-Yetter…