Related papers: Exploiting the Structure in Tensor Decompositions …
In this paper, a numerical method is proposed for canonical polyadic (CP) decomposition of small size tensors. The focus is primarily on decomposition of tensors that correspond to small matrix multiplications. Here, rank of the tensors is…
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…
We provide a computational framework for approximating a class of structured matrices; here, the term structure is very general, and may refer to a regular sparsity pattern (e.g., block-banded), or be more highly structured (e.g., symmetric…
In this paper, we suggest a new method for a given tensor to find CP decompositions using a less number of rank $1$ tensors. The main ingredient is the Least Absolute Shrinkage and Selection Operator (LASSO) by considering the decomposition…
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…
For various $2\leq n,m \leq 6$, we propose some new algorithms for multiplying an $n\times m$ matrix with an $m \times 6$ matrix over a possibly noncommutative coefficient ring.
An algorithm for matrix factorization of polynomials was proposed in \cite{fomatati2022tensor} and it was shown that this algorithm produces better results than the standard method for factoring polynomials on the class of summand-reducible…
We propose a new numerical algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint. Reformulating this computational problem as a system…
Robust tensor CP decomposition involves decomposing a tensor into low rank and sparse components. We propose a novel non-convex iterative algorithm with guaranteed recovery. It alternates between low-rank CP decomposition through gradient…
Low rank tensor decompositions are a powerful tool for learning generative models, and uniqueness results give them a significant advantage over matrix decomposition methods. However, tensors pose significant algorithmic challenges and…
Many recent tensor network algorithms apply unitary operators to parts of a tensor network in order to reduce entanglement. However, many of the previously used iterative algorithms to minimize entanglement can be slow. We introduce an…
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…
Fast matrix multiplication is one of the most fundamental problems in algorithm research. The exponent of the optimal time complexity of matrix multiplication is usually denoted by $\omega$. This paper discusses new ideas for improving the…
An important building block in all current asymptotically fast algorithms for matrix multiplication are tensors with low border rank, that is, tensors whose border rank is equal or very close to their size. To find new asymptotically fast…
This is the second in a series of papers on rank decompositions of the matrix multiplication tensor. We present new rank $23$ decompositions for the $3\times 3$ matrix multiplication tensor $M_{\langle 3\rangle}$. All our decompositions…
Tensor decomposition is a fundamental method used in various areas to deal with high-dimensional data. \emph{Tensor power method} (TPM) is one of the widely-used techniques in the decomposition of tensors. This paper presents a novel tensor…
We propose a strategy for the generation of fast and accurate versions of non-commutative recursive matrix multiplication algorithms. To generate these algorithms, we consider matrix and tensor norm bounds governing the stability and…
In this work, we present a new way to compute the Taylor polynomial of the matrix exponential which reduces the number of matrix multiplications in comparison with the de-facto standard Patterson-Stockmeyer method. This reduction is…
Multilinear and tensor decompositions are a popular tool in linear and multilinear algebra and have a wide range of important applications to modern computing. Our paper of 1972 presented the first nontrivial application of such…
Tensors are a natural way to express correlations among many physical variables, but storing tensors in a computer naively requires memory which scales exponentially in the rank of the tensor. This is not optimal, as the required memory is…