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Composite materials often exhibit mechanical anisotropy owing to the material properties or geometrical configurations of the microstructure. This makes their inverse design a two-fold problem. First, we must learn the type and orientation…

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…

Tucker decomposition is the cornerstone of modern machine learning on tensorial data analysis, which have attracted considerable attention for multiway feature extraction, compressive sensing, and tensor completion. The most challenging…

Ensemble learning is a method that leverages weak learners to produce a strong learner. However, obtaining a large number of base learners requires substantial time and computational resources. Therefore, it is meaningful to study how to…

Tensor models play an increasingly prominent role in many fields, notably in machine learning. In several applications, such as community detection, topic modeling and Gaussian mixture learning, one must estimate a low-rank signal from a…

I propose a straightforward generalization of the projection scheme for elastic tensors introduced by Moakher and Norris [J. Elasticity 85, 215 (2006)] that takes into account also rotations. The "closest" tensor of any desired symmetry to…

We propose a procedure for automated implicit inductive theorem proving for equational specifications made of rewrite rules with conditions and constraints. The constraints are interpreted over constructor terms (representing data values),…

Statistical learning methods show great promise in providing an accurate prediction of materials and molecular properties, while minimizing the need for computationally demanding electronic structure calculations. The accuracy and…

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…

This work investigates a two-stage method for constructing projection-based reduced-order models (ROMs) of parameterized partial differential equations (PDEs). Based on established tensorial ROM methodology, the proposed approach reduces…

In this study, we present a tensor--train framework for nonintrusive operator inference aimed at learning discrete operators and using them to predict solutions of physical governing equations. Our framework comprises three approaches:…

The Nystr\"om method offers an effective way to obtain low-rank approximation of SPD matrices, and has been recently extended and analyzed to nonsymmetric matrices (leading to the generalized Nystr\"om method). It is a randomized,…

The paper addresses the problem of finding the necessary and sufficient conditions to be satisfied by the engineering moduli of an anisotropic material for the elastic energy to be positive for each state of strain or stress. The problem is…

We define a general product of two $n$-dimensional tensors $\mathbb {A}$ and $\mathbb {B}$ with orders $m\ge 2$ and $k\ge 1$, respectively. This product is a generalization of the usual matrix product, and satisfies the associative law.…

Machine-learning techniques have proved successful in identifying ordered phases of matter. However, it remains an open question how far they can contribute to the understanding of phases without broken symmetry, such as spin liquids. Here…

By a tensor we mean an element of a tensor product of vector spaces over a field. Up to a choice of bases in factors of tensor products, every tensor may be coordinatized, that is, represented as an array consisting of numbers. This note is…

A uniform strategy to derive metric tensors in two spatial dimension for interpolation errors and their gradients in $L^p$ norm is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in corresponding metric space, with…

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…

We give a new, constructive uniqueness theorem for tensor decomposition. It applies to order 3 tensors of format $n \times n \times p$ and can prove uniqueness of decomposition for generic tensors up to rank $r=4n/3$ as soon as $p \geq 4$.…

In this manuscript, we introduce the tensor-train reduced basis method, a novel projection-based reduced-order model designed for the efficient solution of parameterized partial differential equations. While reduced-order models are widely…