Kaczmarz algorithm is an efficient iterative algorithm to solve overdetermined consistent system of linear equations. During each updating step, Kaczmarz chooses a hyperplane based on an individual equation and projects the current estimate for the exact solution onto that space to get a new estimate. Many vairants of Kaczmarz algorithms are proposed on how to choose better hyperplanes. Using the property of randomly sampled data in high-dimensional space, we propose an accelerated algorithm based on clustering information to improve block Kaczmarz and Kaczmarz via Johnson-Lindenstrauss lemma. Additionally, we theoretically demonstrate convergence improvement on block Kaczmarz algorithm.
Our work mainly relies on RKA-JL algorithm @cite_9 and RKA-Block algorithm @cite_13 . We will review these algorithms below.
@cite_9 @cite_13
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We develop algorithms and implementations for computing rank-revealing QR (RRQR) factorizations of dense matrices. First, we develop an efficient block algorithm for approximating an RRQR factorization, employing a windowed version of the commonly used Golub pivoting strategy, aided by incremental condition estimation. Second, we develop efficiently implementable variants of guaranteed reliable RRQR algorithms for triangular matrices originally suggested by Chandrasekaran and Ipsen and by Pan and Tang. We suggest algorithmic improvements with respect to condition estimation, termination criteria, and Givens updating. By combining the block algorithm with one of the triangular postprocessing steps, we arrive at an efficient and reliable algorithm for computing an RRQR factorization of a dense matrix. Experimental results on IBM RS 6000 SGI R8000 platforms show that this approach performs up to three times faster that the less reliable QR factorization with column pivoting as it is currently implemented in LAPACK, and comes within 15 of the performance of the LAPACK block algorithm for computing a QR factorization without any column exchanges. Thus, we expect this routine to be useful in may circumstances where numerical rank deficiency cannot be ruled out, but currently has been ignored because of the computational cost of dealing with it. List of Algorithms Preface 1. Introduction. Brief Overview of the State of the Art Notation Review of Relevant Linear Algebra Part I. Krylov Subspace Approximations. 2. Some Iteration Methods. Simple Iteration Orthomin(1) and Steepest Descent Orthomin(2) and CG Orthodir, MINRES, and GMRES Derivation of MINRES and CG from the Lanczos Algorithm 3. Error Bounds for CG, MINRES, and GMRES. Hermitian Problems-CG and MINRES Non-Hermitian Problems-GMRES 4. Effects of Finite Precision Arithmetic. Some Numerical Examples The Lanczos Algorithm A Hypothetical MINRES CG Implementation A Matrix Completion Problem Orthogonal Polynomials 5. BiCG and Related Methods. The Two-Sided Lanczos Algorithm The Biconjugate Gradient Algorithm The Quasi-Minimal Residual Algorithm Relation Between BiCG and QMR The Conjugate Gradient Squared Algorithm The BiCGSTAB Algorithm Which Method Should I Use? 6. Is There A Short Recurrence for a Near-Optimal Approximation? The Faber and Manteuffel Result Implications 7. Miscellaneous Issues. Symmetrizing the Problem Error Estimation and Stopping Criteria Attainable Accuracy Multiple Right-Hand Sides and Block Methods Computer Implementation Part II. Preconditioners. 8. Overview and Preconditioned Algorithms. 9. Two Example Problems. The Diffusion Equation The Transport Equation 10. Comparison of Preconditioners. Jacobi, Gauss--Seidel, SOR The Perron--Frobenius Theorem Comparison of Regular Splittings Regular Splittings Used with the CG Algorithm Optimal Diagonal and Block Diagonal Preconditioners 11. Incomplete Decompositions. Incomplete Cholesky Decomposition Modified Incomplete Cholesky Decomposition 12. Multigrid and Domain Decomposition Methods. Multigrid Methods Basic Ideas of Domain Decomposition Methods. Abstract In this paper, we investigate the convergence of the proximal iteratively reweighted algorithm for a class of nonconvex and nonsmooth problems. Such problems actually include numerous models in the area of signal processing and machine learning research. Two extensions of the algorithm are also studied. We provide a unified scheme for these three algorithms. With the Kurdyka–Łojasiewicz property, we prove that the unified algorithm globally converges to a critical point of the objective function. Given a real matrix A∈Rm×n of rank r, and an integer k<r, the sum of the outer products of top k singular vectors scaled by the corresponding singular values provide the best rank-k approximation Ak to A. When the columns of A have specific meaning, it might be desirable to find good approximations to Ak which use a small number of columns of A. This paper provides a simple greedy algorithm for this problem in Frobenius norm, with guarantees on the performance and the number of columns chosen. The algorithm selects c columns from A with c=O(klogkϵ2η2(A)) such that ‖A−ΠCA‖F≤(1+ϵ)‖A−Ak‖F, where C is the matrix composed of the c columns, ΠC is the matrix projecting the columns of A onto the space spanned by C and η(A) is a measure related to the coherence in the normalized columns of A. The algorithm is quite intuitive and is obtained by combining a greedy solution to the generalization of the well known sparse approximation problem and an existence result on the possibility of sparse approximation. We provide empirical results on various specially constructed matrices comparing our algorithm with the previous deterministic approaches based on QR factorizations and a recently proposed randomized algorithm. The results indicate that in practice, the performance of the algorithm can be significantly better than the bounds suggest. Every day researchers from all over the world have to filter the huge mass of existing research papers with the crucial aim of finding out useful publications related to their current work. In this paper we propose a research paper recommending algorithm based on the Citation Graph and random-walker properties. The PaperRank algorithm is able to assign a preference score to a set of documents contained in a digital library and linked one each other by bibliographic references. A data set of papers extracted by ACM Portal has been used for testing and very promising performances have been measured. We describe and analyze an online algorithm for supervised learning of pseudo-metrics. The algorithm receives pairs of instances and predicts their similarity according to a pseudo-metric. The pseudo-metrics we use are quadratic forms parameterized by positive semi-definite matrices. The core of the algorithm is an update rule that is based on successive projections onto the positive semi-definite cone and onto half-space constraints imposed by the examples. We describe an efficient procedure for performing these projections, derive a worst case mistake bound on the similarity predictions, and discuss a dual version of the algorithm in which it is simple to incorporate kernel operators. The online algorithm also serves as a building block for deriving a large-margin batch algorithm. We demonstrate the merits of the proposed approach by conducting experiments on MNIST dataset and on document filtering. This work focuses on the decoding algorithm of the LDPC large block FEC codes for the packet erasure channel, also called AL-FEC (Application-Level Forward Error Correction). More specifically this work details the design and the performance of a hybrid decoding scheme, that starts with the Zyablov iterative decoding algorithm, a rapid but suboptimal algorithm in terms of erasure recovery capabilities, and, when required, continues with a Gaussian elimination algorithm. For practical reasons this work focuses on two LDPC codes for the erasure channel, namely LDPC-staircase and LDPC-triangle codes. Nevertheless the decoding scheme proposed can be used with other LDPC codes without any problem. The performance experiments carried out show that the erasure recovery capabilities of LDPC-triangle codes are now extremely close to that of an ideal code, even with small block sizes. This is all the more true with small code rates: whereas the Zyablov iterative decoding scheme becomes unusable as the code rate decreases, the Gaussian elimination makes the LDPC-triangle codes almost ideal. In all the tests, when carefully implemented, the LDPC-triangle codec featuring the proposed decoding scheme is fast, and in particular always significantly faster than the reference Reed-Solomon on GF( @math ) codec. The erasure recovery capabilities of LDPC-staircase codes are also significantly improved, even if they remain a little bit farther from an ideal code. Nevertheless, a great advantage is the fact that LDPC-staircase codes remain significantly faster than LDPC-triangle codes, which, for instance, enables their use with larger blocks. All these results make these codes extremely attractive for many situations and contradict the common belief that using Gaussian elimination is not usable because of a prohibitive processing load. Moreover the proposed approach offers an important flexibility in practice, and depending on the situation, one can either choose to favor erasure recovery capabilities or the processing time. In this paper, we consider the approximation of dense block Toeplitz-plus-Hankel matrices by sums of Kronecker products of Toeplitz-plus-Hankel matrices. We present an algorithm for efficiently computing the matrix approximation that requires the factorization of matrices of much smaller dimension than that of the original. The main results are described for block Toeplitz matrices with Toeplitz-plus-Hankel blocks, but the algorithms can be readily adjusted for other related structures that arise in image processing applications, such as block Toeplitz with Toeplitz blocks and block Toeplitz-plus-Hankel with Toeplitz-plus-Hankel blocks. Our work extends the techniques in Kamm and Nagy (SIAM J. Matrix Anal. Appl. 2000; 22:155–172) and (SIAM J. Matrix Anal. Appl. 2004; 25:829–841), which consider similar matrices, but with the added restriction that the matrices have a banded block-banded structure. We illustrate the effectiveness of our algorithm by using the output of the algorithm to construct preconditioners for systems from two different applications: diffuse optical tomography and atmospheric image deblurring. Copyright © 2007 John Wiley & Sons, Ltd. Many optimizing compilers use basic block reordering to reduce conditional branch misprediction penalties, decrease the number of unconditional branches and to improve the instruction cache performance. To perform this optimization one has to solve an NP-hard problem of finding a covering of a weighted control flow graph with a node-disjoint set of simple paths with maximal total weight; the problem is usually solved heuristically. We present the precise method which decomposes the initial graph into the set of subgraphs of certain kind — hammocks — and then solves a subproblem for each subgraph by an exponential branch-and-bound algorithm. The algorithm was tested on several real-world C programs, and its running time was as small as the rest of the compilation time for 91 of the tested graphs. Unfortunately it still failed to finish in a reasonable time for the remaining graphs since they didn’t have enough hammocks. We consider fixing up this drawback as a future work. We study the convergence properties of an alternating proximal minimization algorithm for nonconvex structured functions of the type: L(x,y)=f(x)+Q(x,y)+g(y), where f and g are proper lower semicontinuous functions, defined on Euclidean spaces, and Q is a smooth function that couples the variables x and y. The algorithm can be viewed as a proximal regularization of the usual Gauss-Seidel method to minimize L. We work in a nonconvex setting, just assuming that the function L satisfies the Kurdyka-Łojasiewicz inequality. An entire section illustrates the relevancy of such an assumption by giving examples ranging from semialgebraic geometry to “metrically regular” problems. Our main result can be stated as follows: If L has the Kurdyka-Łojasiewicz property, then each bounded sequence generated by the algorithm converges to a critical point of L. This result is completed by the study of the convergence rate of the algorithm, which depends on the geometrical properties of the function L around its critical points. When specialized to @math and to f, g indicator functions, the algorithm is an alternating projection mehod (a variant of von Neumann's) that converges for a wide class of sets including semialgebraic and tame sets, transverse smooth manifolds or sets with “regular” intersection. To illustrate our results with concrete problems, we provide a convergent proximal reweighted l1 algorithm for compressive sensing and an application to rank reduction problems. Given a dictionary that consists of multiple blocks and a signal that lives in the range space of only a few blocks, we study the problem of finding a block-sparse representation of the signal, i.e., a representation that uses the minimum number of blocks. Motivated by signal image processing and computer vision applications, such as face recognition, we consider the block-sparse recovery problem in the case where the number of atoms in each block is arbitrary, possibly much larger than the dimension of the underlying subspace. To find a block-sparse representation of a signal, we propose two classes of nonconvex optimization programs, which aim to minimize the number of nonzero coefficient blocks and the number of nonzero reconstructed vectors from the blocks, respectively. Since both classes of problems are NP-hard, we propose convex relaxations and derive conditions under which each class of the convex programs is equivalent to the original nonconvex formulation. Our conditions depend on the notions of mutual and cumulative subspace coherence of a dictionary, which are natural generalizations of existing notions of mutual and cumulative coherence. We evaluate the performance of the proposed convex programs through simulations as well as real experiments on face recognition. We show that treating the face recognition problem as a block-sparse recovery problem improves the state-of-the-art results by 10 with only 25 of the training data. Block transform coded images usually suffer from annoying artifacts at low bit rates, caused by the coarse quantization of transform coefficients. In this paper, we propose a new method to reduce compression artifacts by the overlapped-block transform coefficient estimation from non-local blocks. In the proposed method, the discrete cosine transform coefficients of each block are estimated by adaptively fusing two prediction values based on their reliabilities. One prediction is the quantized values of coefficients decoded from the compressed bitstream, whose reliability is determined by quantization steps. The other prediction is the weighted average of the coefficients in nonlocal blocks, whose reliability depends on the variance of the coefficients in these blocks. The weights are used to distinguish the effectiveness of the coefficients in nonlocal blocks to predict original coefficients and are determined by block similarity in transform domain. To solve the optimization problem, the overlapped blocks are divided into several subsets. Each subset contains nonoverlapped blocks covering the whole image and is optimized independently. Therefore, the overall optimization is reduced to a set of sub-optimization problems, which can be easily solved. Finally, we provide a strategy for parameter selection based on the compression levels. Experimental results show that the proposed method can remarkably reduce compression artifacts and significantly improve both the subjective and objective qualities of block transform coded images. We describe new data structures for full and packed storage of dense symmetric triangular arrays that generalize both. Using the new data structures one is led to several new algorithms that save "half" the storage for symmetric matrices and outperform the current blocked based level 3 algorithms in LAPACK. We concentrate on the simplest forms of the new algorithms and show they are a direct generalization of LINPACK. This means that level 3 BLAS's are not required to obtain level 3 performance. The replacement for Level 3 BLAS are so-called kernel routines, see [1], and on IBM platforms they are producible from simple textbook type codes, by the XLF Fortran compiler. In the sequel I will label these "vanilla" codes. On Power3 with a peak performance of 800 MFlops, the results for Cholesky factorization at order n ? 200 is over 720 MFlops and then reaches 735 MFlops at n = 400. Using conventional full format LAPACK DPOTRF with ESSL BLAS's one first gets to 600 MFlops at n ? 600 and only reaches a peak of 620 MFlops. For this result we used simple square blocked full matrix data formats where the blocks themselves are stored in column major (Fortran) order or row major (C) order. The simple algorithms of LU factorization with partial pivoting for this new data format is a direct generalization of LINPACK algorithm DGEFA. Again, no conventional level 3 BLAS's are required; the replacements are again so-called kernel routines. Programming for squared blocked full matrix format can be accomplished in standard Fortran through the use of three and four dimensional arrays. Thus, no new compiler support is necessary. Also we mention that other more complicated algorithms are possible; e.g., recursive ones. The recursive algorithms are also easily programmed via the use of tables that address where the blocks are stored in the two dimensional recursive block array. Finally, we describe block hybrid formats. Doing so allows one to use no additional storage over conventional (full and packed) matrix storage. This means the new algorithms are completely portable. The iteration complexity of the block-coordinate descent (BCD) type algorithm has been under extensive investigation. It was recently shown that for convex problems the classical cyclic BCGD (block coordinate gradient descent) achieves an O(1 r) complexity (r is the number of passes of all blocks). However, such bounds are at least linearly depend on K (the number of variable blocks), and are at least K times worse than those of the gradient descent (GD) and proximal gradient (PG) methods. In this paper, we close such theoretical performance gap between cyclic BCD and GD PG. First we show that for a family of quadratic nonsmooth problems, the complexity bounds for cyclic Block Coordinate Proximal Gradient (BCPG), a popular variant of BCD, can match those of the GD PG in terms of dependency on K (up to a log2 (K) factor). Second, we establish an improved complexity bound for Coordinate Gradient Descent (CGD) for general convex problems which can match that of GD in certain scenarios. Our bounds are sharper than the known bounds as they are always at least K times worse than GD. Our analyses do not depend on the update order of block variables inside each cycle, thus our results also apply to BCD methods with random permutation (random sampling without replacement, another popular variant). In this letter, we propose a new decoding algorithm based on a soft-input soft-output run-length limited (RLL) decoder in visible light communication (VLC) with on–off keying (OOK) modulation and Reed–Solomon (RS) codes. Conventional RLL codes are used for dimming adjustments in VLC; however, in our receiver model, the posterior probability matrix is produced by the new RLL decoder, and it can be matched as the input of the soft RS decoder to enhance the throughput in the VLC system. A significant advantage of our proposed method is that it leads to the enhancement of bit error rate (BER) and frame error rate (FER) performances without any change in the transmitter in VLC systems. The simulation results show that the majority of the BER and FER performances of our proposed method are better than those of hard-input hard-output RLL decoding and soft-input hard-output RLL decoding methods. Given a point q, a reverse k nearest neighbor (RkNN) query retrieves all the data points that have q as one of their k nearest neighbors. Existing methods for processing such queries have at least one of the following deficiencies: (i) they do not support arbitrary values of k (ii) they cannot deal efficiently with database updates, (iii) they are applicable only to 2D data (but not to higher dimensionality), and (iv) they retrieve only approximate results. Motivated by these shortcomings, we develop algorithms for exact processing of RkNN with arbitrary values of k on dynamic multidimensional datasets. Our methods utilize a conventional data-partitioning index on the dataset and do not require any pre-computation. In addition to their flexibility, we experimentally verify that the proposed algorithms outperform the existing ones even in their restricted focus. The nonsmooth and nonconvex regularization has many applications in imaging science and machine learning research due to its excellent recovery performance. A proximal iteratively reweighted nuclear norm algorithm has been proposed for the nonsmooth and nonconvex matrix minimizations. In this paper, we aim to investigate the convergence of the algorithm. With the Kurdyka–Łojasiewicz property, we prove the algorithm globally converges to a critical point of the objective function. The numerical results presented in this paper coincide with our theoretical findings. This paper proposes a method to combine text-based and citation-based retrieval methods in the invalidity patent search. Using the NTCIR-6 test collection including eight years of USPTO patents, we show the effectiveness of our method experimentally. In this paper, we propose a simple, fast decoding algorithm that fosters diversity in neural generation. The algorithm modifies the standard beam search algorithm by adding an inter-sibling ranking penalty, favoring choosing hypotheses from diverse parents. We evaluate the proposed model on the tasks of dialogue response generation, abstractive summarization and machine translation. We find that diverse decoding helps across all tasks, especially those for which reranking is needed. We further propose a variation that is capable of automatically adjusting its diversity decoding rates for different inputs using reinforcement learning (RL). We observe a further performance boost from this RL technique. This paper includes material from the unpublished script "Mutual Information and Diverse Decoding Improve Neural Machine Translation" (Li and Jurafsky, 2016). In this paper, we study online algorithms that schedule malleable jobs, i.e., jobs that can be parallelized on any subset of the available @math m identical machines. We study a model that accounts for the tradeoff between multiprocessor speedup and overhead time, namely, if job @math j has processing requirement @math pj and is assigned to run on @math kj machines, then @math j's execution time becomes @math pj kj+(kj-1)c, where @math c is a constant parameter to the problem. For @math m=2, we present an online algorithm OCS that has a strong competitive ratio of 3 2, matching a previously established lower bound. We also present an online algorithm ASYM2 that is asymptotically @math ((4-∈) (3-∈))-competitive when @math m=2, where @math 0<∈≤2 is a parameter to the algorithm, improving upon an existing asymptotically (3 2)-competitive algorithm. Finally, we present an online algorithm OTO that is strongly @math 2-competitive when @math m=3, improving upon the previous best upper bound of @math 9 4.