In this paper we propose an approach to approximate a truncated singular value decomposition of a large structured matrix. By first decomposing the matrix into a sum of Kronecker products, our approach can be used to approximate a large number of singular values and vectors more efficiently than other well known schemes, such as randomized matrix algorithms or iterative algorithms based on Golub-Kahan bidiagonalization. We provide theoretical results and numerical experiments to demonstrate the accuracy of our approximation and show how the approximation can be used to solve large scale ill-posed inverse problems, either as an approximate filtering method, or as a preconditioner to accelerate iterative algorithms.
The time complexity of the randomized algorithm for computing a truncated SVD approximation with an @math PSF is @math , while the complexity of our method is @math (see formula 6.2 of @cite_19 and to convert notation). In our experience, @math to be effective, and @math may be chosen small enough so as to be roughly constant. Based on these considerations, our method has a practically faster time complexity.
@cite_19
1554201860 2951200405 2139224857 2154318861
Given a matrix A e ℝ m ×n of rank r, and an integer k < r, the top k singular vectors provide the best rank-k approximation to A. When the columns of A have specific meaning, it is desirable to find (provably) "good" approximations to A k which use only a small number of columns in A. Proposed solutions to this problem have thus far focused on randomized algorithms. Our main result is a simple greedy deterministic algorithm with guarantees on the performance and the number of columns chosen. Specifically, our greedy algorithm chooses c columns from A with @math such that @math where C gr is the matrix composed of the c columns, @math is the pseudo-inverse of C gr ( @math is the best reconstruction of A from C gr), and ¼(A) is a measure of the coherence in the normalized columns of A. The running time of the algorithm is O(SVD(A k) + mnc) where SVD(A k) is the running time complexity of computing the first k singular vectors of A. To the best of our knowledge, this is the first deterministic algorithm with performance guarantees on the number of columns and a (1 + e) approximation ratio in Frobenius norm. The algorithm is quite simple and intuitive and is obtained by combining a generalization of the well known sparse approximation problem from information theory with an existence result on the possibility of sparse approximation. Tightening the analysis along either of these two dimensions would yield improved results. We give a randomized @math -time and space algorithm for solving the Shortest Vector Problem (SVP) on n-dimensional Euclidean lattices. This improves on the previous fastest algorithm: the deterministic @math -time and @math -space algorithm of Micciancio and Voulgaris (STOC 2010, SIAM J. Comp. 2013). In fact, we give a conceptually simple algorithm that solves the (in our opinion, even more interesting) problem of discrete Gaussian sampling (DGS). More specifically, we show how to sample @math vectors from the discrete Gaussian distribution at any parameter in @math time and space. (Prior work only solved DGS for very large parameters.) Our SVP result then follows from a natural reduction from SVP to DGS. We also show that our DGS algorithm implies a @math -time algorithm that approximates the Closest Vector Problem to within a factor of @math . In addition, we give a more refined algorithm for DGS above the so-called smoothing parameter of the lattice, which can generate @math discrete Gaussian samples in just @math time and space. Among other things, this implies a @math -time and space algorithm for @math -approximate decision SVP. It is an extreme challenge to produce a nonlinear SVM classifier on very large scale data. In this paper we describe a novel P-packSVM algorithm that can solve the Support Vector Machine (SVM) optimization problem with an arbitrary kernel. This algorithm embraces the best known stochastic gradient descent method to optimize the primal objective, and has 1 ? dependency in complexity to obtain a solution of optimization error ?. The algorithm can be highly parallelized with a special packing strategy, and experiences sub-linear speed-up with hundreds of processors. We demonstrate that P-packSVM achieves accuracy sufficiently close to that of SVM-light, and overwhelms the state-of-the-art parallel SVM trainer PSVM in both accuracy and efficiency. As an illustration, our algorithm trains CCAT dataset with 800k samples in 13 minutes and 95 accuracy, while PSVM needs 5 hours but only has 92 accuracy. We at last demonstrate the capability of P-packSVM on 8 million training samples. Let A be a Las Vegas algorithm, i.e., A is a randomized algorithm that always produces the correct answer when its stops but whose running time is a random variable. The authors consider the problem of minimizing the expected time required to obtain an answer from A using strategies which simulate A as follows: run A for a fixed amount of time t sub 1 , then run A independent for a fixed amount of time t sub 2 , etc. The simulation stops if A completes its execution during any of the runs. Let S=(t sub 1 , t sub 2 ,. . .) be a strategy, and let l sub A =inf sub S T(A,S), where T(A,S) is the expected value of the running time of the simulation of A under strategy S. The authors describe a simple universal strategy S sup univ , with the property that, for any algorithm A, T(A,S sup univ )=O(l sub A log(l sub A )). Furthermore, they show that this is the best performance that can be achieved, up to a constant factor, by any universal strategy. >