Popular deep neural networks (DNNs) spend the majority of their execution time computing convolutions. The Winograd family of algorithms can greatly reduce the number of arithmetic operations required and is present in many DNN software frameworks. However, the performance gain is at the expense of a reduction in floating point (FP) numerical accuracy. In this paper, we analyse the worst case FP error and prove the estimation of norm and conditioning of the algorithm. We show that the bound grows exponentially with the size of the convolution, but the error bound of the algorithm is smaller than the original one. We propose several methods for reducing FP error. We propose a canonical evaluation ordering based on Huffman coding that reduces summation error. We study the selection of sampling "points" experimentally and find empirically good points for the most important sizes. We identify the main factors associated with good points. In addition, we explore other methods to reduce FP error, including mixed-precision convolution, and pairwise summation across DNN channels. Using our methods we can significantly reduce FP error for a given block size, which allows larger block sizes and reduced computation.
To the best of our knowledge, this paper is the first that presents a theoretical analysis of the numerical error of Toom-Cook convolution. We demonstrate that the algorithm is unstable because the properties of algorithm parameters matrices we use ( @math , @math , @math ). We formulate the boundaries of the floating point errors for @math and @math dimensional kernels and inputs so we can resonable choose what we should focus on to improve the accuracy. We formulated the error boundaries for Toom-Cook convolution using similar techniqes to those used for another bilinear problem: the fast matrix multiplication, error estimation by Bini and Lotti @cite_1 , @cite_15 and @cite_9 . We show that algorithm is unstable and how the error depends on each component.
@cite_9 @cite_15 @cite_1
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We perform forward error analysis for a large class of recursive matrix multiplication algorithms in the spirit of Bini and Lotti [Numer. Math. 36:63–72, 1980]. As a consequence of our analysis, we show that the exponent of matrix multiplication (the optimal running time) can be achieved by numerically stable algorithms. We also show that new group-theoretic algorithms proposed in Cohn and Umans [Foundations of Computer Science, 44th Annual IEEE Symposium, pp. 438–449, 2003] and [Foundations of Computer Science, 46th Annual IEEE Symposium, pp. 379–388, 2005] are all included in the class of algorithms to which our analysis applies, and are therefore numerically stable. We perform detailed error analysis for three specific fast group-theoretic algorithms. This paper introduces a parallel and distributed algorithm for solving the following minimization problem with linear constraints: @math minimizef1(x1)+ź+fN(xN)subject toA1x1+ź+ANxN=c,x1źX1,ź,xNźXN,where @math Nź2, @math fi are convex functions, @math Ai are matrices, and @math Xi are feasible sets for variable @math xi. Our algorithm extends the alternating direction method of multipliers (ADMM) and decomposes the original problem into N smaller subproblems and solves them in parallel at each iteration. This paper shows that the classic ADMM can be extended to the N-block Jacobi fashion and preserve convergence in the following two cases: (i) matrices @math Ai are mutually near-orthogonal and have full column-rank, or (ii) proximal terms are added to the N subproblems (but without any assumption on matrices @math Ai). In the latter case, certain proximal terms can let the subproblem be solved in more flexible and efficient ways. We show that @math źxk+1-xkźM2 converges at a rate of o(1 k) where M is a symmetric positive semi-definte matrix. Since the parameters used in the convergence analysis are conservative, we introduce a strategy for automatically tuning the parameters to substantially accelerate our algorithm in practice. We implemented our algorithm (for the case ii above) on Amazon EC2 and tested it on basis pursuit problems with >300 GB of distributed data. This is the first time that successfully solving a compressive sensing problem of such a large scale is reported. We propose a new nonconforming finite element algorithm to approximate the solution to the elliptic problem involving the fractional Laplacian. We first derive an integral representation of the bilinear form corresponding to the variational problem. The numerical approximation of the action of the corresponding stiffness matrix consists of three steps: (i) apply a sinc quadrature scheme to approximate the integral representation by a finite sum where each term involves the solution of an elliptic partial differential equation defined on the entire space, (ii) truncate each elliptic problem to a bounded domain, (iii) use the finite element method for the space approximation on each truncated domain. The consistency error analysis for the three steps is discussed together with the numerical implementation of the entire algorithm. The results of computations are given illustrating the error behavior in terms of the mesh size of the physical domain, the domain truncation parameter and the quadrature spacing parameter. Given @math noisy samples with @math dimensions, where @math , we show that the multi-step thresholding procedure based on the Lasso -- we call it the Thresholded Lasso , can accurately estimate a sparse vector @math in a linear model @math , where @math is a design matrix normalized to have column @math norm @math , and @math . We show that under the restricted eigenvalue (RE) condition (Bickel-Ritov-Tsybakov 09), it is possible to achieve the @math loss within a logarithmic factor of the ideal mean square error one would achieve with an oracle while selecting a sufficiently sparse model -- hence achieving sparse oracle inequalities ; the oracle would supply perfect information about which coordinates are non-zero and which are above the noise level. In some sense, the Thresholded Lasso recovers the choices that would have been made by the @math penalized least squares estimators, in that it selects a sufficiently sparse model without sacrificing the accuracy in estimating @math and in predicting @math . We also show for the Gauss-Dantzig selector (Cand e s-Tao 07), if @math obeys a uniform uncertainty principle and if the true parameter is sufficiently sparse, one will achieve the sparse oracle inequalities as above, while allowing at most @math irrelevant variables in the model in the worst case, where @math is the smallest integer such that for @math , @math . Our simulation results on the Thresholded Lasso match our theoretical analysis excellently. Many data analysis applications deal with large matrices and involve approximating the matrix using a small number of “components.” Typically, these components are linear combinations of the rows and columns of the matrix, and are thus difficult to interpret in terms of the original features of the input data. In this paper, we propose and study matrix approximations that are explicitly expressed in terms of a small number of columns and or rows of the data matrix, and thereby more amenable to interpretation in terms of the original data. Our main algorithmic results are two randomized algorithms which take as input an @math matrix @math and a rank parameter @math . In our first algorithm, @math is chosen, and we let @math , where @math is the Moore-Penrose generalized inverse of @math . In our second algorithm @math , @math , @math are chosen, and we let @math . ( @math and @math are matrices that consist of actual columns and rows, respectively, of @math , and @math is a generalized inverse of their intersection.) For each algorithm, we show that with probability at least @math , @math , where @math is the “best” rank- @math approximation provided by truncating the SVD of @math , and where @math is the Frobenius norm of the matrix @math . The number of columns of @math and rows of @math is a low-degree polynomial in @math , @math , and @math . Both the Numerical Linear Algebra community and the Theoretical Computer Science community have studied variants of these matrix decompositions over the last ten years. However, our two algorithms are the first polynomial time algorithms for such low-rank matrix approximations that come with relative-error guarantees; previously, in some cases, it was not even known whether such matrix decompositions exist. Both of our algorithms are simple and they take time of the order needed to approximately compute the top @math singular vectors of @math . The technical crux of our analysis is a novel, intuitive sampling method we introduce in this paper called “subspace sampling.” In subspace sampling, the sampling probabilities depend on the Euclidean norms of the rows of the top singular vectors. This allows us to obtain provable relative-error guarantees by deconvoluting “subspace” information and “size-of- @math ” information in the input matrix. This technique is likely to be useful for other matrix approximation and data analysis problems. In this paper, we study matrix scaling and balancing, which are fundamental problems in scientific computing, with a long line of work on them that dates back to the 1960s. We provide algorithms for both these problems that, ignoring logarithmic factors involving the dimension of the input matrix and the size of its entries, both run in time @math where @math is the amount of error we are willing to tolerate. Here, @math represents the ratio between the largest and the smallest entries of the optimal scalings. This implies that our algorithms run in nearly-linear time whenever @math is quasi-polynomial, which includes, in particular, the case of strictly positive matrices. We complement our results by providing a separate algorithm that uses an interior-point method and runs in time @math . In order to establish these results, we develop a new second-order optimization framework that enables us to treat both problems in a unified and principled manner. This framework identifies a certain generalization of linear system solving that we can use to efficiently minimize a broad class of functions, which we call second-order robust. We then show that in the context of the specific functions capturing matrix scaling and balancing, we can leverage and generalize the work on Laplacian system solving to make the algorithms obtained via this framework very efficient. A hierarchical Bayes Gaussian Mixture Model (HBGMM) with adaptive sparsity is proposed.We impose our hierarchical Bayes prior on the entries of precision matrices in GMM to encourage sparsity.Our adaptive sparsity can reduce the dimensionality of the parameters in GMM to be estimated.Our HBGMM method does not involve any hyperparameters to be estimated, i.e., it is hyperparameter-free. Gaussian Mixture Model (GMM) has been widely used in statistics for its great flexibility. However, parameter estimation for GMM with high dimensionality is a challenge because of the large number of parameters and the lack of observation data. In this paper, we propose an effective method named hierarchical Bayes based Adaptive Sparsity in Gaussian Mixture Model (ASGMM) to estimate the parameters in a GMM by incorporating a two-layer hierarchical Bayes based adaptive sparsity prior. The prior we impose on the precision matrices can encourage sparsity and hence reduce the dimensionality of the parameters to be estimated. In contrast to the l 1 -norm penalty or Laplace prior, our approach does not involve any hyperparameters that must be tuned, and the sparsity adapts to the observation data. The proposed method is achieved by three steps: first, we formulate an adaptive hierarchical Bayes model of the precision matrices in the GMM with a Jeffrey's noninformative hyperprior, which expresses scale-invariance and, more importantly, is hyperparameter-free and unbiased. Second, we perform a Cholesky decomposition on the precision matrices to impose the positive definite property. Finally, we exploit the expectation maximization (EM) algorithm to obtain the final estimated parameters in the GMM. Experimental results on synthetic and real-world datasets demonstrate that ASGMM cannot only adapt the sparsity of high-dimensional data with small estimated error, but also achieve better clustering performance comparing with several classical methods. In this paper, a novel algorithm to blindly separate an instantaneous linear underdetermined mixture of nonstationary sources is proposed. It means that the number of sources exceeds the number of channels of the available data. The separation is based on the working assumption that the sources are piecewise stationary with a different variance in each block. It proceeds in two steps: 1) estimating the mixing matrix, and 2) computing the optimum beamformer in each block to maximize the signal-to-interference ratio of each separated signal with respect to the remaining signals. Estimating the mixing matrix is accomplished through a specialized tensor decomposition of the set of sample covariance matrices of the received mixture in each block. It utilizes optimum weighting, which allows statistically efficient (CRB attaining) estimation provided that the data obey the assumed Gaussian piecewise stationary model. In simulations, performance of the algorithm is successfully tested on blind separation of 16 speech signals from nine linear instantaneous mixtures of these signals. Recovering low-rank and sparse matrices from incomplete or corrupted observations is an important problem in statistics, machine learning, computer vision, as well as signal and image processing. In theory, this problem can be solved by the natural convex joint mixed relaxations (i.e., l 1 -norm and trace norm) under certain conditions. However, all current provable algorithms suffer from superlinear per-iteration cost, which severely limits their applicability to large-scale problems. In this paper, we propose a scalable, provable and structured robust bilinear factorization (RBF) method to recover low-rank and sparse matrices from missing and grossly corrupted data, i.e., robust matrix completion (RMC), or incomplete and grossly corrupted measurements, i.e., compressive principal component pursuit (CPCP). Specifically, we first present two small-scale matrix trace norm regularized bilinear factorization models for RMC and CPCP problems, in which repetitively calculating SVD of a large-scale matrix is replaced by updating two much smaller factor matrices. Then, we apply the alternating direction method of multipliers (ADMM) to efficiently solve the RMC problems. Finally, we provide the convergence analysis of our algorithm, and extend it to address general CPCP problems. Experimental results verified both the efficiency and effectiveness of our method compared with the state-of-the-art methods. We reconsider randomized algorithms for the low-rank approximation of symmetric positive semi-definite (SPSD) matrices such as Laplacian and kernel matrices that arise in data analysis and machine learning applications. Our main results consist of an empirical evaluation of the performance quality and running time of sampling and projection methods on a diverse suite of SPSD matrices. Our results highlight complementary aspects of sampling versus projection methods; they characterize the effects of common data preprocessing steps on the performance of these algorithms; and they point to important differences between uniform sampling and nonuniform sampling methods based on leverage scores. In addition, our empirical results illustrate that existing theory is so weak that it does not provide even a qualitative guide to practice. Thus, we complement our empirical results with a suite of worst-case theoretical bounds for both random sampling and random projection methods. These bounds are qualitatively superior to existing bounds---e.g. improved additive-error bounds for spectral and Frobenius norm error and relative-error bounds for trace norm error---and they point to future directions to make these algorithms useful in even larger-scale machine learning applications. We reconsider randomized algorithms for the low-rank approximation of symmetric positive semi-definite (SPSD) matrices such as Laplacian and kernel matrices that arise in data analysis and machine learning applications. Our main results consist of an empirical evaluation of the performance quality and running time of sampling and projection methods on a diverse suite of SPSD matrices. Our results highlight complementary aspects of sampling versus projection methods; they characterize the effects of common data preprocessing steps on the performance of these algorithms; and they point to important differences between uniform sampling and nonuniform sampling methods based on leverage scores. In addition, our empirical results illustrate that existing theory is so weak that it does not provide even a qualitative guide to practice. Thus, we complement our empirical results with a suite of worst-case theoretical bounds for both random sampling and random projection methods. These bounds are qualitatively superior to existing bounds--e.g., improved additive-error bounds for spectral and Frobenius norm error and relative-error bounds for trace norm error--and they point to future directions to make these algorithms useful in even larger-scale machine learning applications. We give efficient algorithms for volume sampling, i.e., for picking @math -subsets of the rows of any given matrix with probabilities proportional to the squared volumes of the simplices defined by them and the origin (or the squared volumes of the parallelepipeds defined by these subsets of rows). In other words, we can efficiently sample @math -subsets of @math with probabilities proportional to the corresponding @math by @math principal minors of any given @math by @math positive semi definite matrix. This solves an open problem from the monograph on spectral algorithms by Kannan and Vempala (see Section @math of KV , also implicit in BDM, DRVW ). Our first algorithm for volume sampling @math -subsets of rows from an @math -by- @math matrix runs in @math arithmetic operations (where @math is the exponent of matrix multiplication) and a second variant of it for @math -approximate volume sampling runs in @math arithmetic operations, which is almost linear in the size of the input (i.e., the number of entries) for small @math . Our efficient volume sampling algorithms imply the following results for low-rank matrix approximation: (1) Given @math , in @math arithmetic operations we can find @math of its rows such that projecting onto their span gives a @math -approximation to the matrix of rank @math closest to @math under the Frobenius norm. This improves the @math -approximation of Boutsidis, Drineas and Mahoney BDM and matches the lower bound shown in DRVW . The method of conditional expectations gives a algorithm with the same complexity. The running time can be improved to @math at the cost of losing an extra @math in the approximation factor. (2) The same rows and projection as in the previous point give a @math -approximation to the matrix of rank @math closest to @math under the spectral norm. In this paper, we show an almost matching lower bound of @math , even for @math . Updating probabilistic belief matrices as new observations arrive, in the presence of noise, is a critical part of many algorithms for target tracking in sensor networks. These updates have to be carried out while preserving sum constraints, arising for example, from probabilities. This paper addresses the problem of updating belief matrices to satisfy sum constraints using scaling algorithms. We show that the convergence behavior of the Sinkhorn scaling process, used for scaling belief matrices, can vary dramatically depending on whether the prior unscaled matrix is exactly scalable or only almost scalable. We give an efficient polynomial-time algorithm based on the maximum-flow algorithm that determines whether a given matrix is exactly scalable, thus determining the convergence properties of the Sinkhorn scaling process. We prove that the Sinkhorn scaling process always provides a solution to the problem of minimizing the Kullback-Leibler distance of the physically feasible scaled matrix from the prior constraint-violating matrix, even when the matrices are not exactly scalable. We pose the scaling process as a linearly constrained convex optimization problem, and solve it using an interior-point method. We prove that even in cases in which the matrices are not exactly scalable, the problem can be solved to e-optimality in strongly polynomial time, improving the best known bound for the problem of scaling arbitrary nonnegative rectangular matrices to prescribed row and column sums. This paper studies the estimation of a high-dimensional vector signal where the observation is a known "sparse" linear transformation of the signal corrupted by additive Gaussian noise. A paradigm of such a linear system is code-division multiple access (CDMA) channel with sparse spreading matrix. Assuming a "semi-regular" ensemble of sparse matrix linear transformations, where the bi-partite graph describing the system is asymptotically cycle-free, it is shown that belief propagation (BP) achieves the minimum mean-square error (MMSE) in estimating the transformation of the input vector in the large-system limit. The result holds regardless of the the distribution and power of the input symbols. Furthermore, the mean squared error of estimating each symbol of the input vector using BP is proved to be equal to the MMSE of estimating the same symbol through a scalar Gaussian channel with some degradation in the signal-to-noise ratio (SNR). The degradation, called the efficiency, is determined from a fixed-point equation due to Guo and Verdu, which is a generalization of Tanaka's formula to arbitrary prior distributions. This paper presents a method for using set-based approximations to the Peano-Baker series to compute overapproximations of reachable sets for linear systems with uncertain, time-varying parameters and inputs. Alternative representations for sets of uncertain system matrices are considered, including matrix polytopes, matrix zonotopes, and interval matrices. For each representation, the computational efficiency and resulting approximation error for reachable set computations are evaluated analytically and empirically. As an application, reachable sets are computed for a truck with hybrid dynamics due to a gain-scheduled yaw controller. As an alternative to computing reachable sets for the hybrid model, for which switching introduces an additional overapproximation error, the gain-scheduled controller is approximated with uncertain time-varying parameters, which leads to more efficient and more accurate reachable set computations. In this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the Hermitian complex form. A special case of the problem that we study corresponds to the max-3-cut model used in a recent paper of Goemans and Williamson J. Comput. System Sci., 68 (2004), pp. 442-470]. We first develop a closed-form formula to compute the probability of a complex-valued normally distributed bivariate random vector to be in a given angular region. This formula allows us to compute the expected value of a randomized (with a specific rounding rule) solution based on the optimal solution of the complex semidefinite programming relaxation problem. In particular, we present an @math -approximation algorithm, and then study the limit of that model, in which the problem remains NP-hard. We show that if the objective is to maximize a positive semidefinite Hermitian form, then the randomization-rounding procedure guarantees a worst-case performance ratio of @math , which is better than the ratio of @math for its counterpart in the real case due to Nesterov. Furthermore, if the objective matrix is real-valued positive semidefinite with nonpositive off-diagonal elements, then the performance ratio improves to 0.9349. This contribution is the numerically oriented companion article of the work [E. Cances, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131]. We focus here on the numerical resolution of the embedded corrector problem introduced in [E. Cances, V. Ehrlacher, F. Legoll and B. Stamm, CRAS 2015; E. Cances, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131] in the context of homogenization of diffusion equations. Our approach consists in considering a corrector-type problem, posed on the whole space, but with a diffusion matrix which is constant outside some bounded domain. In [E. Cances, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131], we have shown how to define three approximate homogenized diffusion coefficients on the basis of the embedded corrector problems. We have also proved that these approximations all converge to the exact homogenized coefficients when the size of the bounded domain increases. We show here that, under the assumption that the diffusion matrix is piecewise constant, the corrector problem to solve can be recast as an integral equation. In case of spherical inclusions with isotropic materials, we explain how to efficiently discretize this integral equation using spherical harmonics, and how to use the fast multipole method (FMM) to compute the resulting matrix-vector products at a cost which scales only linearly with respect to the number of inclusions. Numerical tests illustrate the performance of our approach in various settings. We discuss a new robust convergence analysis of the well-known subspace iteration algorithm for computing the dominant singular vectors of a matrix, also known as simultaneous iteration or power method. The result characterizes the convergence behavior of the algorithm when a large amount noise is introduced after each matrix-vector multiplication. While interesting in its own right, the main motivation comes from the problem of privacy-preserving spectral analysis where noise is added in order to achieve the privacy guarantee known as differential privacy. This result leads to nearly tight worst-case bounds for the problem of computing a differentially private low-rank approximation in the spectral norm. Our results extend to privacy-preserving principal component analysis. We obtain improvements for several variants of differential privacy that have been considered. The running time of our algorithm is nearly linear in the input sparsity leading to strong improvements in running time over previous work. Complementing our worst-case bounds, we show that the error dependence of our algorithm on the matrix dimension can be replaced by a tight dependence on the coherence of the matrix. This parameter is always bounded by the matrix dimension but often much smaller. Indeed, the assumption of low coherence is essential in several machine learning and signal processing applications. We observe (Xi,Yi)i=1n where the Yi's are real valued outputs and the Xi's are m × T matrices. We observe a new entry X and we want to predict the output Y associated with it. We focus on the high-dimensional setting, where mT ≫ n. This includes the matrix completion problem with noise, as well as other problems. We consider linear prediction procedures based on different penalizations, involving a mixture of several norms: the nuclear norm, the Frobenius norm and the l1-norm. For these procedures, we prove sharp oracle inequalities, using a statistical learning theory point of view. A surprising fact in our results is that the rates of convergence do not depend on m and T directly. The analysis is conducted without the usually considered incoherency condition on the unknown matrix or restricted isometry condition on the sampling operator. Moreover, our results are the first to give for this problem an analysis of penalization (such as nuclear norm penalization) as a regularization algorithm: our oracle inequalities prove that these procedures have a prediction accuracy close to the deterministic oracle one, given that the reguralization parameters are well-chosen. We present a unified framework for the design and convergence analysis of a class of algorithms based on approximate solution of proximal point subproblems. Our development further enhances the constructive approximation approach of the recently proposed hybrid projection–proximal and extragradient–proximal methods. Specifically, we introduce an even more flexible error tolerance criterion, as well as provide a unified view of these two algorithms. Our general method possesses global convergence and local (super)linear rate of convergence under standard assumptions, while using a constructive approximation criterion suitable for a number of specific implementations. For example, we show that close to a regular solution of a monotone system of semismooth equations, two Newton iterations are sufficient to solve the proximal subproblem within the required error tolerance. Such systems of equations arise naturally when reformulating the nonlinear complementarity problem. *Research of the first author is suppo...