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.