Related papers: Basis-Free Analysis of Singular Tuples and Eigenpa…
Tensor fields depending on other tensor fields are considered. The concept of extended tensor fields is introduced and the theory of differentiation for such fields is developed.
Mnesors are defined as elements of a semimodule over the min-plus integers. This two-sorted structure is able to merge graduation properties of vectors and idempotent properties of boolean numbers, which makes it appropriate for hybrid…
A recent development in random matrix theory, the intrinsic freeness principle, establishes that the spectrum of very general random matrices behaves as that of an associated free operator. This reduces the study of such random matrices to…
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…
This text is a support for different courses of the master of Mechanics of the University Paris-Saclay. The content of this text is an introduction, for graduate students, to tensor algebra and analysis. Far from being exhaustive, the text…
This article introduces an algebraic framework for establishing eigenvalue bounds for symmetric positive definite tensors by leveraging intrinsic invariants, specifically the trace and determinant (resultant). We derive a hierarchy of…
The transversity and the tensor charge of the nucleon, currently under active investigation experimentally and theoretically, are fundamental quantities in hadron physics as well as for our comprehension of the nucleon structure. Some…
Though algebraic geometry over $\mathbb C$ is often used to describe the closure of the tensors of a given size and complex rank, this variety includes tensors of both smaller and larger rank. Here we focus on the $n\times n\times n$…
This paper lies in the intersection of several fields: number theory, lattice theory, multilinear algebra, and scientific computing. We adapt existing solution algorithms for tensor eigenvalue problems to the tensor-train framework. As an…
A \emph{tensor-relational} computation is a relational computation where individual tuples carry vectors, matrices, or higher-dimensional arrays. An advantage of tensor-relational computation is that the overall computation can be executed…
The fundamental matrix and trifocal tensor are convenient algebraic representations of the epipolar geometry of two and three view configurations, respectively. The estimation of these entities is central to most reconstruction algorithms,…
We introduce a new consistency-based approach for defining and solving nonnegative/positive matrix and tensor completion problems. The novelty of the framework is that instead of artificially making the problem well-posed in the form of an…
Tensors are multidimensional analogs of matrices. In this paper, based on degree-theoretic ideas, we study homogeneous nonlinear complementarity problems induced by tensors. By specializing this to $Z$-tensors (which are tensors with…
We present a straightforward analytical-numerical methodology for determining polynomially complete and irreducible scalar-valued invariant sets for anisotropic hyperelasticity. By applying the proposed technique, we obtain irreducible…
It is a significant challenge to design probabilistic programming systems that can accommodate a wide variety of inference strategies within a unified framework. Noting that the versatility of modern automatic differentiation frameworks is…
We introduce the concept of mode-k generalized eigenvalues and eigenvectors of a tensor and prove some properties of such eigenpairs. In particular, we derive an upper bound for the number of equivalence classes of generalized tensor…
We introduce the triangulant of two matrices, and relate it to the existence of orthogonal eigenvectors. We also use it for a new characterization of mutually unbiased bases. Generalizing the notion, we introduce higher order triangulants…
The tubal tensor framework provides a clean and effective algebraic setting for tensor computations, supporting matrix-mimetic features like Singular Value Decomposition and Eckart-Young-like optimality results. Underlying the tubal tensor…
Matrix and tensor completion are frameworks for a wide range of problems, including collaborative filtering, missing data, and image reconstruction. Missing entries are estimated by leveraging an assumption that the matrix or tensor is…
In this paper, we extend some classes of structured matrices to higher order tensors. We discuss their relationships with positive semi-definite tensors and some other structured tensors. We show that every principal sub-tensor of such a…