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arXiv:1807.02513 · Numerical Linear Algebra with Applications

On algorithms for and computing with the tensor ring decomposition

Oscar Mickelin, Sertaç Karaman

Published Feb 24, 2020• 34 citations
arXivHTMLPDF

Abstract

Abstract 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 instead of exponential scaling. In this paper, we investigate even lower storage‐cost representations in the tensor ring format, which is an extension of the tensor train format with variable end‐ranks. Firstly, we introduce two algorithms for converting a tensor in full format to tensor ring format with low storage cost. Secondly, we detail a rounding operation for tensor rings and show how this requires new definitions of common linear algebra operations in the format to obtain storage‐cost savings. Lastly, we introduce algorithms for transforming the graph structure of graph‐based tensor formats, with orders of magnitude lower complexity than existing literature. The efficiency of all algorithms is demonstrated on a number of numerical examples, and in certain cases, we demonstrate significantly higher compression ratios when compared to previous approaches to using the tensor ring format.

DOI
10.1002/nla.2289

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Abstract 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 instead of exponential scaling. In this paper, we investigate even lower storage‐cost representations in the tensor ring format, which is an extension of the tensor train format with variable end‐ranks. Firstly, we introduce two algorithms for converting a tensor in full format to tensor ring format with low storage cost. Secondly, we detail a rounding operation for tensor rings and show how this requires new definitions of common linear algebra operations in the format to obtain storage‐cost savings. Lastly, we introduce algorithms for transforming the graph structure of graph‐based tensor formats, with orders of magnitude lower complexity than existing literature. The efficiency of all algorithms is demonstrated on a number of numerical examples, and in certain cases, we demonstrate significantly higher compression ratios when compared to previous approaches to using the tensor ring format.
Tensor-train (TT) decomposition has gained tremendous popularity due to its efficiency in handling high-dimensional data arising in scientific and quantum computing as well as machine learning applications. It provides a compact representation of matrices and vectors with a Kronecker product-like low-rank structure, and hence enables efficient matrix-vector operations in this compressed form. The vector scalar product is among such key operations, comprising a series of tensor contractions in a specific TT network where the order of contractions significantly impacts the computational cost. In this work, we introduce fast algorithms for finding a near-optimal contraction ordering for such networks. We show that our algorithms find such contraction orderings efficiently, outperforming the ordering methods for general tensor networks. Compared to our approaches, the best existing method incurs a higher cost of up to 15% for xTy, twice for xTAy, and more than ten times for xTABy scalar products, where A and B are matrices expressed in TT format.
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.
We introduce a new SVD-based tensor decomposition method for tensor networks with arbitrary graph topologies, extending classical hierarchical SVD-based techniques to networks with cycles and general connectivity. We also introduce addition and rounding procedures for traceable tensor graphs, enabling step-rounding time integration of high-dimensional PDEs directly in graph format, with rank truncation controlled to a prescribed tolerance at every time step. We demonstrate the new method on the decomposition of multivariate functions and on the numerical solution of the Fokker-Planck equation, and find that the graph-format representation attains comparable or better accuracy than the classical tensor train and hierarchical Tucker tensor formats, while using substantially fewer degrees of freedom at lower computational cost.
The tensor train decomposition decomposes a tensor into a "train" of 3-way tensors that are interconnected through the summation of auxiliary indices. The decomposition is stable, has a well-defined notion of rank and enables the user to perform various linear algebra operations on vectors and matrices of exponential size in a computationally efficient manner. The tensor ring decomposition replaces the train by a ring through the introduction of one additional auxiliary variable. This article discusses a major issue with the tensor ring decomposition: its inability to compute an exact minimal-rank decomposition from a decomposition with sub-optimal ranks. Both the contraction operation and Hadamard product are motivated from applications and it is shown through simple examples how the tensor ring-rounding procedure fails to retrieve minimal-rank decompositions with these operations. These observations, together with the already known issue of not being able to find a best low-rank tensor ring approximation to a given tensor indicate that the applicability of tensor rings is severely limited.
Tensors with unit Frobenius norm are fundamental objects in many fields, including scientific computing and quantum physics, which are able to represent normalized eigenvectors and pure quantum states. While the tensor train decomposition provides a powerful low-rank format for tackling high-dimensional problems, it does not intrinsically enforce the unit-norm constraint. To address this, we introduce the normalized tensor train (NTT) decomposition, which aims to approximate a tensor by unit-norm tensors in tensor train format. The low-rank structure of NTT decomposition not only saves storage and computational cost but also preserves the underlying unit-norm structure. We prove that the set of fixed-rank NTT tensors forms a smooth manifold, and the corresponding Riemannian geometry is derived, paving the way for geometric methods. We propose NTT-based methods for low-rank tensor recovery, high-dimensional eigenvalue problem, estimation of stabilizer rank, and calculation of the minimum output Rényi 2-entropy of quantum channels. Numerical experiments demonstrate the superior efficiency and scalability of the proposed NTT-based methods.
In this paper we review basic and emerging models and associated algorithms for large-scale tensor networks, especially Tensor Train (TT) decompositions using novel mathematical and graphical representations. We discus the concept of tensorization (i.e., creating very high-order tensors from lower-order original data) and super compression of data achieved via quantized tensor train (QTT) networks. The purpose of a tensorization and quantization is to achieve, via low-rank tensor approximations "super" compression, and meaningful, compact representation of structured data. The main objective of this paper is to show how tensor networks can be used to solve a wide class of big data optimization problems (that are far from tractable by classical numerical methods) by applying tensorization and performing all operations using relatively small size matrices and tensors and applying iteratively optimized and approximative tensor contractions. Keywords: Tensor networks, tensor train (TT) decompositions, matrix product states (MPS), matrix product operators (MPO), basic tensor operations, tensorization, distributed representation od data optimization problems for very large-scale problems: generalized eigenvalue decomposition (GEVD), PCA/SVD, canonical correlation analysis (CCA).