Related papers: Solving Tensor Low Cycle Rank Approximation
We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The method combines functional tensor train (FTT) series expansions, operator splitting time integration, and a new…
We show that a generic tensor $T\in \mathbb{F}^{n\times n\times \dots\times n}$ of order $k$ and CP rank $d$ can be uniquely recovered from $n\log n+dn\log \log n +o(n\log \log n) $ uniformly random entries with high probability if $d$ and…
Machine learning and data mining algorithms are becoming increasingly important in analyzing large volume, multi-relational and multi--modal datasets, which are often conveniently represented as multiway arrays or tensors. It is therefore…
Probabilistic context-free grammars (PCFGs) with neural parameterization have been shown to be effective in unsupervised phrase-structure grammar induction. However, due to the cubic computational complexity of PCFG representation and…
We investigate structural complexity measures on digraphs, in particular the cycle rank. This concept is intimately related to a classical topic in formal language theory, namely the star height of regular languages. We explore this…
Low-rank tensor approximation error bounds are proposed for the case of noisy input data that depend on low-rank representation type, rank and the dimensionality of the tensor. The bounds show that high-dimensional low-rank structured…
In this short paper we bridge two seemingly unrelated sparse approximation topics: continuous sparse coding and low-rank approximations. We show that for a specific choice of continuous dictionary, linear systems with nuclear-norm…
Tensor network contractions are widely used in statistical physics, quantum computing, and computer science. We introduce a method to efficiently approximate tensor network contractions using low-rank approximations, where each intermediate…
We address prediction problems on tabular categorical data, where each instance is defined by multiple categorical attributes, each taking values from a finite set. These attributes are often referred to as fields, and their categorical…
The concept of tensor rank, introduced in the twenties, has been popularized at the beginning of the seventies. This has allowed to carry out Factor Analysis on arrays with more than two indices. The generic rank may be seen as an upper…
We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we obtain a rigorous…
A major challenge to the problem of community question answering is the lexical and semantic gap between the sentence representations. Some solutions to minimize this gap includes the introduction of extra parameters to deep models or…
The so-called block-term decomposition (BTD) tensor model, especially in its rank-$(L_r,L_r,1)$ version, has been recently receiving increasing attention due to its enhanced ability of representing systems and signals that are composed of…
The results of Strassen and Raz show that good enough tensor rank lower bounds have implications for algebraic circuit/formula lower bounds. We explore tensor rank lower and upper bounds, focusing on explicit tensors. For odd d, we…
There has been an increased interest in multimodal language processing including multimodal dialog, question answering, sentiment analysis, and speech recognition. However, naturally occurring multimodal data is often imperfect as a result…
The block tensor of trifocal tensors provides crucial geometric information on the three-view geometry of a scene. The underlying synchronization problem seeks to recover camera poses (locations and orientations up to a global…
In this paper, we introduce a new tensor decomposition for third order tensors, which decomposes a third order tensor to three third order low rank tensors in a balanced way. We call such a decomposition the triple decomposition, and the…
The goals of this work are two-fold: firstly, to propose a new theoretical framework for representing random fields on a large class of multidimensional geometrical domain in the tensor train format; secondly, to develop a new algorithm…
We consider the task of low-multilinear-rank functional regression, i.e., learning a low-rank parametric representation of functions from scattered real-valued data. Our first contribution is the development and analysis of an efficient…
Recently, there has been a growing interest in efficient numerical algorithms based on tensor networks and low-rank techniques to approximate high-dimensional functions and solutions to high-dimensional PDEs. In this paper, we propose a new…