Related papers: Tensor Unfolding Characterization
Recent efforts to unravel the mystery of implicit regularization in deep learning have led to a theoretical focus on matrix factorization -- matrix completion via linear neural network. As a step further towards practical deep learning, we…
Random matrices now play a role in many parts of computational mathematics. To advance these applications, it is desirable to have tools that are flexible, easy to use, and powerful. Over the last 25 years, researchers have developed a…
In this paper, the canonical polyadic (CP) decomposition of tensors that corresponds to matrix multiplications is studied. Finding the rank of these tensors and computing the decompositions is a fundamental problem of algebraic complexity…
This paper introduces a new mathematical framework for analysis and optimization of tensor expressions within an enclosing loop. Tensors are multi-dimensional arrays of values. They are common in high performance computing (HPC) and machine…
Tensors decompositions are a class of tools for analysing datasets of high dimensionality and variety in a natural manner, with the Canonical Polyadic Decomposition (CPD) being a main pillar. While the notion of CPD is closely intertwined…
Over recent years it has become well accepted that user interest is not static or immutable. There are a variety of contextual factors, such as time of day, the weather or the user's mood, that influence the current interests of the user.…
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.
We introduce and extend the outer product and contractive product of tensors and matrices, and present some identities in terms of these products. We offer tensor expressions of derivatives of tensors, focus on the tensor forms of…
The embedding layers transforming input words into real vectors are the key components of deep neural networks used in natural language processing. However, when the vocabulary is large, the corresponding weight matrices can be enormous,…
Tensor decompositions such as the canonical format and the tensor train format have been widely utilized to reduce storage costs and operational complexities for high-dimensional data, achieving linear scaling with the input dimension…
Non-negative matrix factorization (NMF) is a fundamental matrix decomposition technique that is used primarily for dimensionality reduction and is increasing in popularity in the biological domain. Although finding a unique NMF is generally…
Since its introduction by Gauss, Matrix Algebra has facilitated understanding of scientific problems, hiding distracting details and finding more elegant and efficient ways of computational solving. Today's largest problems, which often…
In recent years several classes of structured matrices are extended to classes of tensors in the context of tensor complementarity problem. The tensor complementarity problem is a class of nonlinear complementarity problem where the…
In data processing and machine learning, an important challenge is to recover and exploit models that can represent accurately the data. We consider the problem of recovering Gaussian mixture models from datasets. We investigate symmetric…
Tensor factorization arises in many machine learning applications, such knowledge base modeling and parameter estimation in latent variable models. However, numerical methods for tensor factorization have not reached the level of maturity…
Tensor decompositions are promising tools for big data analytics as they bring multiple modes and aspects of data to a unified framework, which allows us to discover complex internal structures and correlations of data. Unfortunately most…
Many problems can be formulated as recovering a low-rank tensor. Although an increasingly common task, tensor recovery remains a challenging problem because of the delicacy associated with the decomposition of higher order tensors. To…
Large amount of multidimensional data represented by multiway arrays or tensors are prevalent in modern applications across various fields such as chemometrics, genomics, physics, psychology, and signal processing. The structural complexity…
Diagonalization, or eigenvalue decomposition, is very useful in many areas of applied mathematics, including signal processing and quantum physics. Matrix decomposition is also a useful tool for approximating matrices as the product of a…
Our goal here is to see the space of matrices of a given size from a geometric and topological perspective, with emphasis on the families of various ranks and how they fit together. We pay special attention to the nearest orthogonal…