Various applications in signal processing and machine learning give rise to highly structured spectral optimization problems characterized by low-rank solutions. Two important examples that motivate this work are optimization problems from phase retrieval and from blind deconvolution, which are designed to yield rank-1 solutions. An algorithm is described that is based on solving a certain constrained eigenvalue optimization problem that corresponds to the gauge dual which, unlike the more typical Lagrange dual, has an especially simple constraint. The dominant cost at each iteration is the computation of rightmost eigenpairs of a Hermitian operator. A range of numerical examples illustrate the scalability of the approach.
Other researchers have recognized the need for algorithms with low per-iteration costs that scale well for large-scale, low-rank spectral optimization problems. Notable efforts include @cite_11 , @cite_18 , and @cite_8 , who advocate variations of the Frank-Wolfe (FW) method to solve some version of the problem where @math is a differentiable function. For example, the choice @math yields a problem related to . The asymmetric version of the problem with @math is easily accommodated by simply replacing the above constraints. For simplicity, here we focus on the symmetric case, though our approach applies equally to the asymmetric case. The main benefit of using FW for this problem is that each iteration requires only a rightmost eigenvalue of the gradient @math , and therefore has the same per-iteration cost of the method that we consider, which requires a rightmost eigenvalue of the same-sized matrix @math . The same Krylov-based eigensolvers apply in both cases.
@cite_8 @cite_18 @cite_11
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We present a method for solving the shortest transshipmen problem - also known as uncapacitated minimum cost flow - up to a multiplicative error of @math in undirected graphs with polynomially bounded integer edge weights using a tailored gradient descent algorithm. An important special case of the transshipment problem is the single-source shortest paths (SSSP) problem. Our gradient descent algorithm takes @math iterations, and in each iteration it needs to solve the transshipment problem up to a multiplicative error of @math , where @math is the number of nodes. In particular, this allows us to perform a single iteration by computing a solution on a sparse spanner of logarithmic stretch. As a consequence, we improve prior work by obtaining the following results: (1) Broadcast congest model: @math -approximate SSSP using @math rounds, where @math is the (hop) diameter of the network. (2) Broadcast congested clique model: @math -approximate transshipment and SSSP using @math rounds. (3) Multipass streaming model: @math -approximate transshipment and SSSP using @math space and @math passes. The previously fastest algorithms for these models leverage sparse hop sets. We bypass the hop set construction; computing a spanner is sufficient with our method. The above bounds assume non-negative integer edge weights that are polynomially bounded in @math ; for general non-negative weights, running times scale with the logarithm of the maximum ratio between non-zero weights. In case of asymmetric costs, running times scale with the maximum ratio between the costs of both directions over all edges. A number of problems in probability and statistics can be addressed using the multivariate normal (Gaussian) distribution. In the one-dimensional case, computing the probability for a given mean and variance simply requires the evaluation of the corresponding Gaussian density. In the @math -dimensional setting, however, it requires the inversion of an @math covariance matrix, @math , as well as the evaluation of its determinant, @math . In many cases, such as regression using Gaussian processes, the covariance matrix is of the form @math , where @math is computed using a specified covariance kernel which depends on the data and additional parameters (hyperparameters). The matrix @math is typically dense, causing standard direct methods for inversion and determinant evaluation to require @math work. This cost is prohibitive for large-scale modeling. Here, we show that for the most commonly used covariance functions, the matrix @math can be hierarchically factored into a product of block low-rank updates of the identity matrix, yielding an @math algorithm for inversion. More importantly, we show that this factorization enables the evaluation of the determinant @math , permitting the direct calculation of probabilities in high dimensions under fairly broad assumptions on the kernel defining @math . Our fast algorithm brings many problems in marginalization and the adaptation of hyperparameters within practical reach using a single CPU core. The combination of nearly optimal scaling in terms of problem size with high-performance computing resources will permit the modeling of previously intractable problems. We illustrate the performance of the scheme on standard covariance kernels. This paper introduces a novel algorithm to approximate the matrix with minimum nuclear norm among all matrices obeying a set of convex constraints. This problem may be understood as the convex relaxation of a rank minimization problem and arises in many important applications as in the task of recovering a large matrix from a small subset of its entries (the famous Netflix problem). Off-the-shelf algorithms such as interior point methods are not directly amenable to large problems of this kind with over a million unknown entries. This paper develops a simple first-order and easy-to-implement algorithm that is extremely efficient at addressing problems in which the optimal solution has low rank. The algorithm is iterative, produces a sequence of matrices @math , and at each step mainly performs a soft-thresholding operation on the singular values of the matrix @math . There are two remarkable features making this attractive for low-rank matrix completion problems. The first is that the soft-thresholding operation is applied to a sparse matrix; the second is that the rank of the iterates @math is empirically nondecreasing. Both these facts allow the algorithm to make use of very minimal storage space and keep the computational cost of each iteration low. On the theoretical side, we provide a convergence analysis showing that the sequence of iterates converges. On the practical side, we provide numerical examples in which @math matrices are recovered in less than a minute on a modest desktop computer. We also demonstrate that our approach is amenable to very large scale problems by recovering matrices of rank about 10 with nearly a billion unknowns from just about 0.4 of their sampled entries. Our methods are connected with the recent literature on linearized Bregman iterations for @math minimization, and we develop a framework in which one can understand these algorithms in terms of well-known Lagrange multiplier algorithms. SUMMARY We investigate approximations to eigenfunctions of a certain class of elliptic operators in Rd by finite sums of products of functions with separated variables and especially conditions providing an exponential decrease of the error with respect to the number of terms. The consistent use of tensor formats can be regarded as a base for a new class of rank-truncated iterative eigensolvers. The computational cost is almost linear in the univariate problem size n, while traditional method scale like nd. Tensor methods can be applied to solving large-scale spectral problems in computational quantum chemistry, for example, the Schrodinger, Hartree–Fock and Kohn–Sham equations in electronic structure calculations. The results of numerical experiments clearly indicate the linear-logarithmic scaling of the low-rank tensor method in n. The algorithms work equally well for the computation of both minimal and maximal eigenvalues of the discrete elliptic operators. Copyright © 2011 John Wiley & Sons, Ltd.