We propose a combined tensor format, which encapsulates the benefits of Tucker, tensor train (TT), and quantized TT (QTT) formats. The structure is composed of subtensors in TT representations, so the approximation problem is proven to be stable. We describe all important algebraic and optimization operations, which are recast to the TT routines. Several examples on explicit function and operator representations are provided. The asymptotic storage complexity is at most cubic in the rank parameter, that is larger than for the global QTT approximation, but the numerical examples manifest that the ranks in the two-level format usually increase more slowly with the approximation accuracy than the QTT ones. In particular, we observe that high rank peaks, which usually occur in the TT/QTT representations, are significantly relaxed. Thus the reduced costs can be achieved.
We propose a combined tensor format, which encapsulates the benefits of Tucker, Tensor Train (TT) and Quantized TT (QTT) formats. The structure is composed of subtensors in TT representations, so the approximation problem is proven to be stable. We describe all important algebraic and optimization operations, which are recast to the TT routines. Several examples on explicit function and operator representations are provided. The asymptotic storage complexity is at most cubic in the rank parameter, that is larger than for the QTT format, but the numerical examples manifest, that the ranks in the two-level format increase usually slower with the approximation accuracy than the QTT ones. In particular, we observe, that high rank peaks, which usually occur in the QTT representation, are significantly relaxed. Thus the reduced costs can be achieved.
Recent achievements in the field of tensor product approximation provide promising new formats for the representation of tensors in form of tree tensor networks. In contrast to the canonical r-term representation (CANDECOMP, PARAFAC), these new formats provide stable representations, while the amount of required data is only slightly larger. The tensor train (TT) format [SIAM J. Sci. Comput., 33 (2011), pp. 2295–2317], a simple special case of the hierarchical Tucker format [J. Fourier Anal. Appl., 5 (2009), p. 706], is a useful prototype for practical low-rank tensor representation. In this article, we show how optimization tasks can be treated in the TT format by a generalization of the well-known alternating least squares (ALS) algorithm and by a modified approach (MALS) that enables dynamical rank adaptation. A formulation of the component equations in terms of so-called retraction operators helps to show that many structural properties of the original problems transfer to the micro-iterations, giving what is to our knowledge the first stable generic algorithm for the treatment of optimization tasks in the tensor format. For the examples of linear equations and eigenvalue equations, we derive concrete working equations for the micro-iteration steps; numerical examples confirm the theoretical results concerning the stability of the TT decomposition and of ALS and MALS but also show that in some cases, high TT ranks are required during the iterative approximation of low-rank tensors, showing some potential of improvement.
We focus on the construction of explicit low-rank representations of matrices in the tensor train (TT) and quantized tensor train (QTT) formats which have been proposed recently for the low-parametric structured representation of large-scale tensors. The matrices under consideration are discretizations of the Laplace operator in a hypercube (in one and many dimensions) and their inverses (in one dimension). For these matrices we derive explicit and exact QTT representations of low QTT ranks independent of the numbers
d and
n2 of dimensions and entries in each dimension. This implies that for the matrices considered the storage cost is
O(dlogn), i.e., logarithmic with respect to the total number
n2d of entries. The same applies to the computational complexity of the QTT-structured operations with these matrices, which now depends on the QTT ranks of the other operands. The general result of the paper is the notation and technique we introduce in order to examine the QTT structure of matrices analytically. They prove to be an efficient tool of studying other tensors related to particular computational problems, which are not considered in this paper.
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 study the tensor structure of two operations: the transformation of a given multidimensional vector into a multilevel Toeplitz matrix and the convolution of two given multidimensional vectors. We show that the low-rank tensor structure of the input is preserved in the output and propose efficient algorithms for these operations in the newly introduced quantized tensor train (QTT) format. Consider a
d-dimensional
2n×⋯×2n-vector
x. If it is represented elementwise, the number of parameters is
(2n)d. However, if we assume that
x is given in a QTT representation with ranks bounded by
p, the number of parameters is reduced to
O(dp2logn). Under this assumption we show how the multilevel Toeplitz matrix generated by
x can be obtained in the QTT format with ranks bounded by
2p in
O(dp2logn) operations. We also describe how the convolution
x⋆y of
x and a
d-dimensional
n×⋯×n-vector
y can be computed in the QTT format with ranks bounded by
2t in
O(dt2logn) operations, provided that the matrix
xy′ is given in the same format with ranks bounded by
t. We exploit an algorithm for the inexact matrix-vector multiplication in the QTT format to accelerate the convolution algorithm dramatically. The performance of our approach to convolution is demonstrated with numerical examples, including the computation of the Newton potential of a strong cusp on fine grids with up to
220×220×220 points in three dimensions.
In the context of big data, high-order tensor decompositions have to face a new challenge in terms of storage and computational costs. The tensor train (TT) decomposition provides a very useful graph-based model reduction, whose storage cost grows linearly with the tensor order
D. The computation of the TT-core tensors and TT-ranks can be done in a stable sequential (i.e., noniterative) way thanks to the popular TT-SVD algorithm. In this paper, we exploit the ideas developed for the hierarchical/tree Tucker decomposition in the context of the TT decomposition. Specifically, a new efficient estimation scheme, called TT-HSVD (Tensor-Train Hierarchical SVD), is proposed as a solution to compute the TT decomposition of a high-order tensor. The new algorithm simultaneously delivers the TT-core tensors and their TT-ranks in a hierarchical way. It is a stable (i.e., noniterative) and computationally more efficient algorithm than TT-SVD, which is very important when dealing with large-scale data. The TT-HSVD algorithm uses a new reshaping strategy and a tailored partial SVD, which allows us to deal with smaller matrices compared to those of the TT-SVD. In addition, TT-HSVD is well suited for a parallel processing architecture. An algebraic analysis of the two algorithms is carried out, showing that TT-SVD and TT-HSVD compute the same TT-ranks and TT-core tensors up to specific bases. Simulation results for different tensor orders and dimensions corroborate the effectiveness of the proposed algorithm.