相关论文: Tensor Codes and their Invariants

Moment invariants are a powerful tool for the generation of rotation-invariant descriptors needed for many applications in pattern detection, classification, and machine learning. A set of invariants is optimal if it is complete,…

With a simple generic approach, we develop a classification that encodes and measures the strength of completeness (or compactness) properties in various types of spaces and ordered structures. The approach also allows us to encode notions…

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…

Finding the rank of a tensor is a problem that has many applications. Unfortunately it is often very difficult to determine the rank of a given tensor. Inspired by the heuristics of convex relaxation, we consider the nuclear norm instead of…

Tensors are fundamental in mathematics, computer science, and physics. Their study through algebraic geometry and representation theory has proved very fruitful in the context of algebraic complexity theory and quantum information. In…

In this work we introduce the concept of a sub-space decomposition, subject to a partition of the coordinates. Considering metrics determined by partial orders in the set of coordinates, the so called poset metrics, we show the existence of…

We propose a simple connection between matrix quantum mechanics and tensor networks. This allows us to imbue tensor networks with some interesting additional structure. The geometry of the graph describing the tensor network state is…

For a growing number of applications such as cellular, peer-to-peer, and sensor networks, efficient error-free transmission of data through a network is essential. Toward this end, K\"{o}tter and Kschischang propose the use of subspace…

Random tensors are the natural generalization of random matrices to higher order objects. They provide generating functions for random geometries and, assuming some familiarity with random matrix theory and quantum field theory, we discuss…

We characterise the sensitivity of several additive tensor decompositions with respect to perturbations of the original tensor. These decompositions include canonical polyadic decompositions, block term decompositions, and sums of tree…

Tensors, or multidimensional arrays, are data structures that can naturally represent visual data of multiple dimensions. Inherently able to efficiently capture structured, latent semantic spaces and high-order interactions, tensors have a…

Tensors play a central role in many modern machine learning and signal processing applications. In such applications, the target tensor is usually of low rank, i.e., can be expressed as a sum of a small number of rank one tensors. This…

Rank modulation is a way of encoding information to correct errors in flash memory devices as well as impulse noise in transmission lines. Modeling rank modulation involves construction of packings of the space of permutations equipped with…

Symmetry properties of r-times covariant tensors T can be described by certain linear subspaces W of the group ring K[S_r] of a symmetric group S_r. If for a class of tensors T such a W is known, the elements of the orthogonal subspace…

In this book, the authors introduce the notion of set codes, set bicodes and set n-codes. These are the most generalized notions of semigroup n-codes and group n-codes. Several types of set n-codes are defined. Several examples are given to…

Tensor network methods have been a key ingredient of advances in condensed matter physics and have recently sparked interest in the machine learning community for their ability to compactly represent very high-dimensional objects. Tensor…

In this paper, the problem of the identification of the symmetry class of a given tensor is asked. Contrary to classical approaches which are based on the spectral properties of the linear operator describing the elasticity, our setting is…

Let $A$ be a finite or countable alphabet and let $\theta$ be literal (anti)morphism onto $A^*$ (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under…

Tensor decomposition plays a key role in identifying common features across a collection of matrices in many areas of science. A fundamental need in big data research is to process data tabulated as large-scale matrices using eigenvectors.…

Differential geometries derived from tensor decompositions have been extensively studied and provided the foundations for a variety of efficient numerical methods. Despite the practical success of the tensor ring (TR) decomposition, its…