Related papers: Tensor-based derivation of standard vector identit…
Following up on a previous analysis of graph embeddings, we generalize and expand some results to the general setting of vector symbolic architectures (VSA) and hyperdimensional computing (HDC). Importantly, we explore the mathematical…
A covariant scalar-tensor-vector gravity theory is developed which allows the gravitational constant $G$, a vector field coupling $\omega$ and the vector field mass $\mu$ to vary with space and time. The equations of motion for a test…
Vector representations of graphs and relational structures, whether hand-crafted feature vectors or learned representations, enable us to apply standard data analysis and machine learning techniques to the structures. A wide range of…
Vector-space models, from word embeddings to neural network parsers, have many advantages for NLP. But how to generalise from fixed-length word vectors to a vector space for arbitrary linguistic structures is still unclear. In this paper we…
Dimensionality reduction is an effective method for learning high-dimensional data, which can provide better understanding of decision boundaries in human-readable low-dimensional subspace. Linear methods, such as principal component…
Tensor networks, a class of variational quantum many-body wave functions have attracted considerable research interest across many disciplines, including classical machine learning. Recently, Aizpurua et al. demonstrated explainable anomaly…
Vector is a physical quantity and it does not depend on any co-ordinate system. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. The expansion in orthonormal basis is…
A spinorial approach to 6-dimensional differential geometry is constructed and used to analyze tensor fields of low rank, with special attention to the Weyl tensor. We perform a study similar to the 4-dimensional case, making full use of…
This paper presents an extension of Correspondence Analysis (CA) to tensors through High Order Singular Value Decomposition (HOSVD) from a geometric viewpoint. Correspondence analysis is a well-known tool, developed from principal component…
We prove that any derived equivalence between derived-discrete algebras of finite global dimension is standard, that is, isomorphic to the derived tensor functor by a two-sided tilting complex.
We consider two hypothesis testing problems for low-rank and high-dimensional tensor signals, namely the tensor signal alignment and tensor signal matching problems. These problems are challenging due to the high dimension of tensors and…
In this paper, we present a partial survey of the tools borrowed from tensor algebra, which have been utilized recently in Statistics and Signal Processing. It is shown why the decompositions well known in linear algebra can hardly be…
The metric-affine gravity provides a useful framework for analyzing gravitational dynamics since it treats metric tensor and affine connection as fundamentally independent variables. In this work, we show that, a metric-affine gravity…
Eigenvectors of tensors, as studied recently in numerical multilinear algebra, correspond to fixed points of self-maps of a projective space. We determine the number of eigenvectors and eigenvalues of a generic tensor, and we show that the…
Eigenvectors of stress-energy tensor (the source in Einstein's equations) form privileged bases in description of the corresponding space-times. When one or more of these vector fields are rotating (the property well determined in…
This paper connects a vector-based composition model to a formal semantics, the Dependency-based Compositional Semantics (DCS). We show theoretical evidence that the vector compositions in our model conform to the logic of DCS.…
A definition is suggested for affine symmetry tensors, which generalize the notion of affine vectors in the same way that (conformal) Killing tensors generalize (conformal) Killing vectors. An identity for these tensors is proved, which…
The Bel-Robinson tensor is analyzed as a linear map on the space of the traceless symmetric tensors. This study leads to an algebraic classification that refines the usual Petrov-Bel classification of the Weyl tensor. The new classes…
Graph convolutional neural networks have been instrumental in machine learning of material properties. When representing tensorial properties, weights and descriptors of a physics-informed network must obey certain transformation rules to…
There is derived, for a conformally flat three-space, a family of linear second-order partial differential operators which send vectors into tracefree, symmetric two-tensors. These maps, which are parametrized by conformal Killing vectors…