Related papers: Tensor network ranks
The tensor train (TT) rank has received increasing attention in tensor completion due to its ability to capture the global correlation of high-order tensors ($\textrm{order} >3$). For third order visual data, direct TT rank minimization has…
Low rank regularization, in essence, involves introducing a low rank or approximately low rank assumption for matrix we aim to learn, which has achieved great success in many fields including machine learning, data mining and computer…
To determine whether an $n\times n$-matrix has rank at most $r$ it suffices to check that the $(r+1)\times (r+1)$-minors have rank at most $r$. In other words, to describe the set of $n\times n$-matrices with the property of having rank at…
The matrix rank and its positive versions are robust for small approximations, i.e. they do not decrease under small perturbations. In contrast, the multipartite tensor rank can collapse for arbitrarily small errors, i.e. there may be a gap…
Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. We discuss how to incorporate a global symmetry, given by a…
The tensor rank decomposition is a useful tool for the geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able…
Despite the omnipresence of tensors and tensor operations in modern deep learning, the use of tensor mathematics to formally design and describe neural networks is still under-explored within the deep learning community. To this end, we…
A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a sum of symmetric outer product of vectors. A rank-1 order-k…
The notion of `stable rank' of a matrix is central to the analysis of randomized matrix algorithms, covariance estimation, deep neural networks, and recommender systems. We compare the properties of the stable rank and intrinsic dimension…
Graph Neural Networks (GNNs) are attracting growing attention due to their effectiveness and flexibility in modeling a variety of graph-structured data. Exiting GNN architectures usually adopt simple pooling operations (eg. sum, average,…
In pursuit of reinforcement learning systems that could train in physical environments, we investigate multi-task approaches as a means to alleviate the need for massive data acquisition. In a tabular scenario where the Q-functions are…
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…
In this lecture note, we discuss a fundamental concept, referred to as the {\it characteristic rank}, which suggests a general framework for characterizing the basic properties of various low-dimensional models used in signal processing.…
Learning relies on coordinated synaptic changes in recurrently connected populations of neurons. Therefore, understanding the collective evolution of synaptic connectivity over learning is a key challenge in neuroscience and machine…
Graph classification is a significant problem in many scientific domains. It addresses tasks such as the classification of proteins and chemical compounds into categories according to their functions, or chemical and structural properties.…
We prove a generalization to Jennrich's uniqueness theorem for tensor decompositions in the undercomplete setting. Our uniqueness theorem is based on an alternative definition of the standard tensor decomposition, which we call…
Recent findings indicate that over-parametrization, while crucial for successfully training deep neural networks, also introduces large amounts of redundancy. Tensor methods have the potential to efficiently parametrize over-complete…
The trace norm is widely used in multi-task learning as it can discover low-rank structures among tasks in terms of model parameters. Nowadays, with the emerging of big datasets and the popularity of deep learning techniques, tensor trace…
Matrix factorization is a well-studied task in machine learning for compactly representing large, noisy data. In our approach, instead of using the traditional concept of matrix rank, we define a new notion of link-rank based on a…
The minimum rank of a graph G is the minimum rank over all real symmetric matrices whose off-diagonal sparsity pattern is the same as that of the adjacency matrix of G. In this note we present the first exact algorithm for the minimum rank…