Related papers: From computation to comparison of tensor decomposi…
Hermitian tensors are natural generalizations of Hermitian matrices, while possessing rather different properties. A Hermitian tensor is separable if it has a Hermitian decomposition with only positive coefficients, i.e., it is a sum of…
A tree decomposition of a graph facilitates computations by grouping vertices into bags that are interconnected in an acyclic structure, hence their importance in a plethora of problems such as query evaluation over databases and inference…
Tensor train decomposition is widely used in machine learning and quantum physics due to its concise representation of high-dimensional tensors, overcoming the curse of dimensionality. Cross approximation-originally developed for…
We investigate the computational complexity of tensor rank, a concept that plays fundamental role in different topics of modern applied mathematics. For tensors over any integral domain, we prove that the rank problem is polynomial time…
Comparing tensors and identifying their (dis)similar structures is fundamental in understanding the underlying phenomena for complex data. Tensor decomposition methods help analysts extract tensors' essential characteristics and aid in…
Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. We discuss how to incorporate a global symmetry, given by a…
Hankel tensors are generalizations of Hankel matrices. This article studies the relations among various ranks of Hankel tensors. We give an algorithm that can compute the Vandermonde ranks and decompositions for all Hankel tensors. For a…
Assume we are given a sum of linear measurements of $s$ different rank-$r$ matrices of the form $y = \sum_{k=1}^{s} \mathcal{A}_k ({X}_k)$. When and under which conditions is it possible to extract (demix) the individual matrices ${X}_k$…
Tensors decompositions are a class of tools for analysing datasets of high dimensionality and variety in a natural manner, with the Canonical Polyadic Decomposition (CPD) being a main pillar. While the notion of CPD is closely intertwined…
Complicated mathematical equations involving products of tensors with permutation symmetries, frequently encountered in fields such as general relativity and quantum chemistry (e.g., equations in high-order coupled cluster theories),…
Tensor completion refers to the task of estimating the missing data from an incomplete measurement or observation, which is a core problem frequently arising from the areas of big data analysis, computer vision, and network engineering. Due…
Dimensionality reduction for high-order tensors is a challenging problem. In conventional approaches, higher order tensors are `vectorized` via Tucker decomposition to obtain lower order tensors. This will destroy the inherent high-order…
Second-order semantic parsing with end-to-end mean-field inference has been shown good performance. In this work we aim to improve this method by modeling label correlations between adjacent arcs. However, direct modeling leads to memory…
The low-rank canonical polyadic tensor decomposition is useful in data analysis and can be computed by solving a sequence of overdetermined least squares subproblems. Motivated by consideration of sparse tensors, we propose sketching each…
We derive and implement a new way of solving coupled cluster equations with lower computational scaling. Our method is based on decomposition of both amplitudes and two electron integrals, using a combination of tensor hypercontraction and…
The problem of partitioning a large and sparse tensor is considered, where the tensor consists of a sequence of adjacency matrices. Theory is developed that is a generalization of spectral graph partitioning. A best rank-$(2,2,\lambda)$…
In this paper, we investigate whether an a priori disambiguation of word senses is strictly necessary or whether the meaning of a word in context can be disambiguated through composition alone. We evaluate the performance of off-the-shelf…
An important conjecture in additive combinatorics, number theory, and algebraic geometry posits that the partition rank and analytic rank of tensors are equal up to a constant, over any finite field. We prove the conjecture up to a…
Tensors are becoming increasingly common in data mining, and consequently, tensor factorizations are becoming more and more important tools for data miners. When the data is binary, it is natural to ask if we can factorize it into binary…
The theory and computation of tensors with different tensor products play increasingly important roles in scientific computing and machine learning. Different products aim to preserve different algebraic properties from the matrix algebra,…