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Register a new userA Krylov-Schur like method for computing the best rank-$(r_1,r_2,r_3)$ approximation of large and sparse tensors
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Added by
Lars Eld\\'en
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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The paper is concerned with methods for computing the best low multilinear rank approximation of large and sparse tensors. Krylov-type methods have been used for this problem; here blo
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The problem of partitioning a large and sparse tensor is considered, where the tensor consists of a sequence of adjacency matrices. Theory is developed that is a generalization of spectral graph partitioning. A best rank-$(2,2,lambda)$ approximation is computed for $lambda=1,2,3$, and the partitioning is computed from the orthogonal matrices and the core tensor of the approximation. It is shown that if the tensor has a certain reducibility structure, then the solution of the best approximation problem exhibits the reducibility structure of the tensor. Further, if the tensor is close to being reducible, then still the solution of the exhibits the structure of the tensor. Numerical examples with synthetic data corroborate the theoretical results. Experiments with tensors from applications show that the method can be used to extract relevant information from large, sparse, and noisy data.
Sparse tensor best rank-1 approximation (BR1Approx), which is a sparsity generalization of the dense tensor BR1Approx, and is a higher-order extension of the sparse matrix BR1Approx, is one of the most important problems in sparse tensor decomposition and related problems arising from statistics and machine learning. By exploiting the multilinearity as well as the sparsity structure of the problem, four approximation algorithms are proposed, which are easily implemented, of low computational complexity, and can serve as initial procedures for iterative algorithms. In addition, theoretically guaranteed worst-case approximation lower bounds are proved for all the algorithms. We provide numerical experiments on synthetic and real data to illustrate the effectiveness of the proposed algorithms.
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 of spectral graph partitioning to tensors, and it gives a reordering of the tensor that clusters the information. The second method gives an expansion of the tensor in sparse rank-(2,2,1) terms, where the terms correspond to graphs. The low-rank approximations are computed using an efficient Krylov-Schur type algorithm that avoids filling in the sparse data. The methods are applied to topic search in news text, a tensor representing conference author-terms-years, and network traffic logs.
This paper introduces new solvers for the computation of low-rank approximate solutions to large-scale linear problems, with a particular focus on the regularization of linear inverse problems. Although Krylov methods incorporating explicit projections onto low-rank subspaces are already used for well-posed systems that arise from discretizing stochastic or time-dependent PDEs, we are mainly concerned with algorithms that solve the so-called nuclear norm regularized problem, where a suitable nuclear norm penalization on the solution is imposed alongside a fit-to-data term expressed in the 2-norm: this has the effect of implicitly enforcing low-rank solutions. By adopting an iteratively reweighted norm approach, the nuclear norm regularized problem is reformulated as a sequence of quadratic problems, which can then be efficiently solved using Krylov methods, giving rise to an inner-outer iteration scheme. Our approach differs from the other solvers available in the literature in that: (a) Kronecker product properties are exploited to define the reweighted 2-norm penalization terms; (b) efficient preconditioned Krylov methods replace gradient (projection) methods; (c) the regularization parameter can be efficiently and adaptively set along the iterations. Furthermore, we reformulate within the framework of flexible Krylov methods both the new inner-outer methods for nuclear norm regularization and some of the existing Krylov methods incorporating low-rank projections. This results in an even more computationally efficient (but heuristic) strategy, that does not rely on an inner-outer iteration scheme. Numerical experiments show that our new solvers are competitive with other state-of-the-art solvers for low-rank problems, and deliver reconstructions of increased quality with respect to other classical Krylov methods.
New algorithms are proposed for the Tucker approximation of a 3-tensor, that access it using only the tensor-by-vector-by-vector multiplication subroutine. In the matrix case, Krylov methods are methods of choice to approximate the dominant column and row subspaces of a sparse or structured matrix given through the matrix-by-vector multiplication subroutine. Using the Wedderburn rank reduction formula, we propose an algorithm of matrix approximation that computes Krylov subspaces and allows generalization to the tensor case. Several variants of proposed tensor algorithms differ by pivoting strategies, overall cost and quality of approximation. By convincing numerical experiments we show that the proposed methods are faster and more accurate than the minimal Krylov recursion, proposed recently by Elden and Savas.
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