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Register a new userWedderburn rank reduction and Krylov subspace method for tensor approximation. Part 1: Tucker case
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Added by
Dmitry Savostyanov V.
Publication date
2010
fields
Informatics Engineering
and research's language is
English
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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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We propose a new method for the approximate solution of the Lyapunov equation with rank-$1$ right-hand side, which is based on extended rational Krylov subspace approximation with adaptively computed shifts. The shift selection is obtained from the connection between the Lyapunov equation, solution of systems of linear ODEs and alternating least squares method for low-rank approximation. The numerical experiments confirm the effectiveness of our approach.
We study the use of Krylov subspace recycling for the solution of a sequence of slowly-changing families of linear systems, where each family consists of shifted linear systems that differ in the coefficient matrix only by multiples of the identity. Our aim is to explore the simultaneous solution of each family of shifted systems within the framework of subspace recycling, using one augmented subspace to extract candidate solutions for all the shifted systems. The ideal method would use the same augmented subspace for all systems and have fixed storage requirements, independent of the number of shifted systems per family. We show that a method satisfying both requirements cannot exist in this framework. As an alternative, we introduce two schemes. One constructs a separate deflation space for each shifted system but solves each family of shifted systems simultaneously. The other builds only one recycled subspace and constructs approximate corrections to the solutions of the shifted systems at each cycle of the iterative linear solver while only minimizing the base system residual. At convergence of the base system solution, we apply the method recursively to the remaining unconverged systems. We present numerical examples involving systems arising in lattice quantum chromodynamics.
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.
We develop K$omega$, an open-source linear algebra library for the shifted Krylov subspace methods. The methods solve a set of shifted linear equations $(z_k I-H)x^{(k)}=b, (k=0,1,2,...)$ for a given matrix $H$ and a vector $b$, simultaneously. The leading order of the operational cost is the same as that for a single equation. The shift invariance of the Krylov subspace is the mathematical foundation of the shifted Krylov subspace methods. Applications in materials science are presented to demonstrate the advantages of the algorithm over the standard Krylov subspace methods such as the Lanczos method. We introduce benchmark calculations of (i) an excited (optical) spectrum and (ii) intermediate eigenvalues by the contour integral on the complex plane. In combination with the quantum lattice solver $mathcal{H} Phi$, K$omega$ can realize parallel computation of excitation spectra and intermediate eigenvalues for various quantum lattice models.
Low-rank tensor recovery problems have been widely studied in many applications of signal processing and machine learning. Tucker decomposition is known as one of the most popular decompositions in the tensor framework. In recent years, researchers have developed many state-of-the-art algorithms to address the problem of low-Tucker-rank tensor recovery. Motivated by the favorable properties of the stochastic algorithms, such as stochastic gradient descent and stochastic iterative hard thresholding, we aim to extend the well-known stochastic iterative hard thresholding algorithm to the tensor framework in order to address the problem of recovering a low-Tucker-rank tensor from its linear measurements. We have also developed linear convergence analysis for the proposed method and conducted a series of experiments with both synthetic and real data to illustrate the performance of the proposed method.
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