Related papers: The Hackbusch conjecture on tensor formats - part …

It has long been conjectured that hypotheses spaces suitable for data that is compositional in nature, such as text or images, may be more efficiently represented with deep hierarchical networks than with shallow ones. Despite the vast…

Modern sensing and metrology systems now stream terabytes of heterogeneous, high-dimensional (HD) data profiles, images, and dense point clouds, whose natural representation is multi-way tensors. Understanding such data requires regression…

Recently proposed budding tree is a decision tree algorithm in which every node is part internal node and part leaf. This allows representing every decision tree in a continuous parameter space, and therefore a budding tree can be jointly…

Construction of phylogenetic trees and networks for extant species from their characters represents one of the key problems in phylogenomics. While solution to this problem is not always uniquely defined and there exist multiple methods for…

Due to the multi-linearity of tensors, most algorithms for tensor optimization problems are designed based on the block coordinate descent method. Such algorithms are widely employed by practitioners for their implementability and…

We establish basic information about border rank algorithms for the matrix multiplication tensor and other tensors with symmetry. We prove that border rank algorithms for tensors with symmetry (such as matrix multiplication and the…

Information is extracted from large and sparse data sets organized as 3-mode tensors. Two methods are described, based on best rank-(2,2,2) and rank-(2,2,1) approximation of the tensor. The first method can be considered as a generalization…

The so-called block-term decomposition (BTD) tensor model has been recently receiving increasing attention due to its enhanced ability of representing systems and signals that are composed of \emph{blocks} of rank higher than one, a…

In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given tensor by a tensor that is sparsely representable. For matrices, i.e. 2-tensors, such a representation can be obtained…

Tree tensor networks, or tree-based tensor formats, are prominent model classes for the approximation of high-dimensional functions in computational and data science. They correspond to sum-product neural networks with a sparse connectivity…

Kruskal's theorem states that a sum of product tensors constitutes a unique tensor rank decomposition if the so-called k-ranks of the product tensors are large. In this work, we propose a conjecture in which the k-rank condition of…

Convolutional neural networks show outstanding results in a variety of computer vision tasks. However, a neural network architecture design usually faces a trade-off between model performance and computational/memory complexity. For some…

A tensor network is a type of decomposition used to express and approximate large arrays of data. A given data-set, quantum state or higher dimensional multi-linear map is factored and approximated by a composition of smaller multi-linear…

Tensor network contraction is central to problems ranging from many-body physics to computer science. We describe how to approximate tensor network contraction through bond compression on arbitrary graphs. In particular, we introduce a…

Neural networks (NNs) and decision trees (DTs) are both popular models of machine learning, yet coming with mutually exclusive advantages and limitations. To bring the best of the two worlds, a variety of approaches are proposed to…

We consider the NP-hard Tree Containment problem that has important applications in phylogenetics. The problem asks if a given leaf-labeled network contains a subdivision of a given leaf-labeled tree. We develop a fast algorithm for the…

This work is concerned with tree tensor network operators (TTNOs) for representing quantum Hamiltonians. We first establish a mathematical framework connecting tree topologies with state diagrams. Based on these, we devise an algorithm for…

Tensor is the most basic and essential data structure of nowadays artificial intelligence (AI) system. The natural properties of Tensor, especially the memory-continuity and slice-independence, make it feasible for training system to…

Tensor type data are becoming important recently in various application fields. We determine a rank of a tensor T so that A+T is diagonalizable for a given 3-tensor A with 2 slices over the complex and real number field.

This paper proposes a novel approach to tensor completion, which recovers missing entries of data represented by tensors. The approach is based on the tensor train (TT) rank, which is able to capture hidden information from tensors thanks…