Related papers: An OpenMath Content Dictionary for Tensor Concepts
We analyse perception and memory, using mathematical models for knowledge graphs and tensors, to gain insights into the corresponding functionalities of the human mind. Our discussion is based on the concept of propositional sentences…
We show that standard identities and theorems for lattice models with $U(1)$ symmetry get re-expressed discretely in the tensorial formulation of these models. We explain the geometrical analogy between the continuous lattice equations of…
Recent advances in {matrix-mimetic} tensor frameworks have made it possible to preserve linear algebraic properties for multilinear data analysis and, as a result, to obtain optimal representations of multiway data. Matrix mimeticity arises…
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…
The representation theory of tensor functions is essential to constitutive modeling of materials including both mechanical and physical behaviors. Generally, material symmetry is incorporated in the tensor functions through a structural or…
We develop algebraic methods for computations with tensor data. We give 3 applications: extracting features that are invariant under the orthogonal symmetries in each of the modes, approximation of the tensor spectral norm, and…
Music is an art, perceived in unique ways by every listener, coming from acoustic signals. In the meantime, standards as musical scores exist to describe it. Even if humans can make this transcription, it is costly in terms of time and…
By consireding representation theory for non-associative algebras we construct the fundamental and adjoint representations of the octonion algebra. We then show how these representations by associative matrices allow a consistent octonionic…
These lecture notes are intended as an introduction to several notions of tensor rank and their connections to the asymptotic complexity of matrix multiplication. The latter is studied with the exponent of matrix multiplication, which will…
Multipartite quantum scenarios are a significant and challenging resource in quantum information science. Tensors provide a powerful framework for representing multipartite quantum systems. In this work, we introduce the role of…
The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated…
We study pairs of associative algebras and linear functionals. New results together with corrected proofs for previously published material are presented. In particular, we prove the identity ind Mat(n) (#) A=n * ind A$ for…
The full exploitation of non-abelian symmetries in tensor network states (TNS) derived from a given lattice Hamiltonian is highly attractive in various aspects. From a theoretical perspective, it can offer deep insights into the…
The big data era is swamping areas including data analysis, machine/deep learning, signal processing, statistics, scientific computing, and cloud computing. The multidimensional feature and huge volume of big data put urgent requirements to…
Tensor models are measures for random tensors. They generalise matrix models and were developed to study random geometry in arbitrary dimension. Moreover, they are strongly connected to quantum gravity theories as additionally to the…
We propose neural-symbolic integration for abstract concept explanation and interactive learning. Neural-symbolic integration and explanation allow users and domain-experts to learn about the data-driven decision making process of large…
The main concepts of the theory of tensors are presented. The emphasis is on the basic notions of tensor algebra and practical skills in culculations involving the Kronecker delta and Levi-Civita symbol. Sixty routine exercises are…
In many-particle problems involving interacting fermions or bosons, the most natural language for expressing the Hamiltonian, the observables, and the basis states is the language of the second-quantization operators. It thus appears…
The classification of all fourth-order anisotropic tensor classes for classical linear elasticity is well known. In this article, we review the related problem of explicitly computing the dimension and the expressions of the elements…
In this paper we study the computation of both algebraic and non-algebraic tensor functions under the tensor-tensor multiplication with linear maps. In the case of algebraic tensor functions, we prove that the asymptotic exponent of both…