Related papers: From computation to comparison of tensor decomposi…

Tensor operations are surging as the computational building blocks for a variety of scientific simulations and the development of high-performance kernels for such operations is known to be a challenging task. While for operations on one-…

Symmetric tensor operations arise in a wide variety of computations. However, the benefits of exploiting symmetry in order to reduce storage and computation is in conflict with a desire to simplify memory access patterns. In this paper, we…

The orthogonal decomposition factorizes a tensor into a sum of an orthogonal list of rankone tensors. We present several properties of orthogonal rank. We find that a subtensor may have a larger orthogonal rank than the whole tensor and…

We prove that the partition rank and the analytic rank of tensors are equal up to a constant, over finite fields of any characteristic and any large enough cardinality depending on the analytic rank. Moreover, we show that a plausible…

Conjugate partial-symmetric (CPS) tensors are the high-order generalization of Hermitian matrices. As the role played by Hermitian matrices in matrix theory and quadratic optimization, CPS tensors have shown growing interest recently in…

A theorem of J. Kruskal from 1977, motivated by a latent-class statistical model, established that under certain explicit conditions the expression of a 3-dimensional tensor as the sum of rank-1 tensors is essentially unique. We give a new…

Subtracting a critical rank-one approximation from a matrix always results in a matrix with a lower rank. This is not true for tensors in general. Motivated by this, we ask the question: what is the closure of the set of those tensors for…

Over recent years it has become well accepted that user interest is not static or immutable. There are a variety of contextual factors, such as time of day, the weather or the user's mood, that influence the current interests of the user.…

In the last decades, tensors have emerged as the right tool to represent multidimensional data in a compact yet informative manner. Moreover, it is well-known that by performing low-rank factorizations of such tensors one is often able to…

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…

Causal inference from observational data often assumes "ignorability," that all confounders are observed. This assumption is standard yet untestable. However, many scientific studies involve multiple causes, different variables whose…

Higher-order tensors appear in various areas of mechanics as well as physics, medicine or earth sciences. As these tensors are highly complex, most are not well understood. Thus, the analysis and the visualization process form a highly…

There is a significant expansion in both volume and range of applications along with the concomitant increase in the variety of data sources. These ever-expanding trends have highlighted the necessity for more versatile analysis tools that…

We present an iterative algorithm, called the symmetric tensor eigen-rank-one iterative decomposition (STEROID), for decomposing a symmetric tensor into a real linear combination of symmetric rank-1 unit-norm outer factors using only…

We introduce an elementary method to study the border rank of polynomials and tensors, analogous to the apolarity lemma. This can be used to describe the border rank of all cases uniformly, including those very special ones that resisted a…

We study the symmetric tensor rank of multiplication over finite field extensions using linearized polynomials. Via field trace, symmetric linearized polynomials are identified with symmetric bilinear forms and symmetric matrices, allowing…

Because of the attractiveness of the canonical polyadic (CP) tensor decomposition in various applications, several algorithms have been designed to compute it, but efficient ones are still lacking. Iterative deflation algorithms based on…

It has been shown that a best rank-R approximation of an order-k tensor may not exist when R>1 and k>2. This poses a serious problem to data analysts using tensor decompositions. It has been observed numerically that, generally, this issue…

In problems involving approximation, completion, denoising, dimension reduction, estimation, interpolation, modeling, order reduction, regression, etc, we argue that the near-universal practice of assuming that a function, matrix, or tensor…

We develop a systematic way to solve linear equations involving tensors of arbitrary rank. We start off with the case of a rank $3$ tensor, which appears in many applications, and after finding the condition for a unique solution we derive…