Related papers: Demystifying Tubal Tensor Algebra
When designing new materials, it is often necessary to tailor the material design (with respect to its design parameters) to have some desired properties (e.g. Young's modulus). As the set of design parameters grow, the search space grows…
In the mid thirties Murray and von Neumann found a natural way to associate a von Neumann algebra $L(\Gamma)$ to any countable discrete group $\Gamma$. Classifying $L(\Gamma)$ in term of $\Gamma$ is a notoriously complex problem as in…
We show that a particular class of parallel algorithm for linear functions can be straightforwardly generalized to a parallel algorithm of their tensor product. The central idea is to take a model of parallel algorithms -- Bulk Synchronous…
Twisted generalized Weyl algebras (TGWAs) $A(R,\sigma,t)$ are defined over a base ring $R$ by parameters $\sigma$ and $t$, where $\sigma$ is an $n$-tuple of automorphisms, and $t$ is an $n$-tuple of elements in the center of $R$. We show…
Casting neural networks in generative frameworks is a highly sought-after endeavor these days. Contemporary methods, such as Generative Adversarial Networks, capture some of the generative capabilities, but not all. In particular, they lack…
Modeling of multidimensional signal using tensor is more convincing than representing it as a collection of matrices. The tensor based approaches can explore the abundant spatial and temporal structures of the mutlidimensional signal. The…
Category theory provides a means through which many far-ranging fields of mathematics can be related by their similar structure. In a paper by Robinson [2], this interconnectivity afforded by categorical perspectives allowed for the…
In this paper, we mainly develop the well-known vector and matrix polynomial extrapolation methods in tensor framework. To this end, some new products between tensors are defined and the concept of positive definitiveness is extended for…
This is the second part of the paper (the first part is published in Jour. of AMS, vol.9, 1135--1170, q-alg/9508017). In the first part, we defined for every modular tensor category (MTC) inner products on the spaces of morphisms and proved…
This paper is concerned with the prime spectrum of a tensor product of algebras over a field. It seeks necessary and sufficient conditions for such a tensor product to have the S-property, strong S-property, and catenarity. Its main results…
This paper proposes a novel paradigm for machine learning that moves beyond traditional parameter optimization. Unlike conventional approaches that search for optimal parameters within a fixed geometric space, our core idea is to treat the…
Interpretability for machine learning models is becoming more and more important as machine learning models become more complex. The functional ANOVA model, which decomposes a high-dimensional function into a sum of lower dimensional…
Matrix and tensor-guided parametrization for Natural Language Processing (NLP) models is fundamentally useful for the improvement of the model's systematic efficiency. However, the internal links between these two algebra structures and…
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…
Tensor contraction operations in computational chemistry consume significant fractions of computing time on large-scale computing platforms. The widespread use of tensor contractions between large multi-dimensional tensors in describing…
Motivated by the study of dynamic control systems, this paper proposes novel algebraic operations on cubic matrices to construct both linear and nonlinear controlled dynamics. The standard t-product of cubic matrices imposes strict…
Due to the explosive growth of large-scale data sets, tensors have been a vital tool to analyze and process high-dimensional data. Different from the matrix case, tensor decomposition has been defined in various formats, which can be…
We construct the tensor product for f-algebras, including proving a universal property for it, and investigate how it preserves algebraic properties of the factors.
In this paper, we investigate whether the tensor product of two frames, each individually localised with respect to a spectral matrix algebra, is also localised with respect to a suitably chosen tensor product algebra. We provide a partial…
This paper introduces Multidimensional Task Learning (MTL), a unified mathematical framework based on Generalized Einstein MLPs (GE-MLPs) that operate directly on tensors via the Einstein product. We argue that current computer vision task…