Tensor Train Decomposition

Chang Nie, Huan Wang, Le Tian

Jincheng Zhang

Relational tensor decomposition aims to uncover the underlying relationships between entities represented in a multi-dimensional tensor. Traditional methods often struggle with scalability and interpretability, particularly when dealing with complex relational structures. This paper presents a novel approach to relational tensor decomposition that leverages the efficiency of Tensor Trains (TTs) combined with a sparsity-inducing constraint based on relational dependencies. We decompose relational tensors using TT decomposition, and then introduce a sparsity pattern that prioritizes connections between related entities. This process effectively builds a sparse TT representation that explicitly encodes the relational structure of the data. The resulting method demonstrates improved efficiency and enhanced interpretability compared to standard TT decomposition, offering a promising solution for analyzing complex relational data. The core claim is improved efficiency and interpretability of relational tensor decomposition by sparsifying the Tensor Train representation based on relational dependencies. The core mechanism is the application of Tensor Train decomposition to relational tensors, coupled with a sparsity constraint prioritizing connections between related entities.

Timo Felser, Simone Montangero

Huazhong Liu, Laurence Tianruo Yang, Yimu Guo +2 more

Cyber-physical-social big data generated from ubiquitous devices and diverse spaces generally are multi-source, heterogeneous, and deeply intertwined. To efficiently analyze and handle the ubiquitous cyber-physical-social big data, tensor is considered as an effective tool, but the curse of dimensionality is still the main bottleneck of tensor-based big data analysis. Tensor networks can considerably alleviate or overcome it through the tensor approximate theory. Therefore, this paper focuses on developing an efficient big data processing framework based on tensor networks and providing an incremental tensor train decomposition approach for the streaming big data. Concretely, this paper first presents a hierarchical cyber-physical-social big data processing framework composed of three planes, namely, data representation and decomposition, data storage and processing, and data analysis and service, in which tensor train (TT) and quantized TT decompositions are particularly introduced to remarkably overcome the curse of dimensionality. Besides, to efficiently handle the continuous streaming big data and avoid the repeated decomposition for the history data, an incremental tensor train decomposition (ITTD) approach is proposed and the complexities are further analyzed in detail. Experimental results demonstrate that ITTD demonstrably outperforms the nonincremental TT decomposition in execution time on the precise of guaranteeing the nearly equal approximation error.

Simone Montangero

Namgil Lee, Andrzej S Cichocki

We discuss extended definitions of linear and multilinear operations such as Kronecker, Hadamard, and contracted products, and establish links between them for tensor calculus. Then we introduce effective low-rank tensor approximation techniques including Candecomp/Parafac (CP), Tucker, and tensor train (TT) decompositions with a number of mathematical and graphical representations. We also provide a brief review of mathematical properties of the TT decomposition as a low-rank approximation technique. With the aim of breaking the curse-of-dimensionality in large-scale numerical analysis, we describe basic operations on large-scale vectors, matrices, and high-order tensors represented by TT decomposition. The proposed representations can be used for describing numerical methods based on TT decomposition for solving large-scale optimization problems such as systems of linear equations and symmetric eigenvalue problems.