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2. An Alternating Manifold Proximal Gradient Method for Sparse Principal Component Analysis and Sparse Canonical Correlation Analysis

   https://doi.org/10.1287/ijoo.2019.0032  

   Chen, Shixiang ; Ma, Shiqian ; Xue, Lingzhou ; Zou, Hui ( January 2020 , INFORMS Journal on Optimization)

   Sparse principal component analysis and sparse canonical correlation analysis are two essential techniques from high-dimensional statistics and machine learning for analyzing large-scale data. Both problems can be formulated as an optimization problem with nonsmooth objective and nonconvex constraints. Because nonsmoothness and nonconvexity bring numerical difficulties, most algorithms suggested in the literature either solve some relaxations of them or are heuristic and lack convergence guarantees. In this paper, we propose a new alternating manifold proximal gradient method to solve these two high-dimensional problems and provide a unified convergence analysis. Numerical experimental results are reported to demonstrate the advantages of our algorithm. 

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3. Learning Mixtures of Sparse Linear Regressions Using Sparse Graph Codes

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4. Algorithm 1021: SPEX Left LU, Exactly Solving Sparse Linear Systems via a Sparse Left-looking Integer-preserving LU Factorization

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   Lourenco, Christopher ; Chen, Jinhao ; Moreno-Centeno, Erick ; Davis, Timothy A. ( June 2022 , ACM Transactions on Mathematical Software)

   SPEX Left LU is a software package for exactly solving unsymmetric sparse linear systems. As a component of the sparse exact (SPEX) software package, SPEX Left LU can be applied to any input matrix, A , whose entries are integral, rational, or decimal, and provides a solution to the system \( Ax = b \) , which is either exact or accurate to user-specified precision. SPEX Left LU preorders the matrix A with a user-specified fill-reducing ordering and computes a left-looking LU factorization with the special property that each operation used to compute the L and U matrices is integral. Notable additional applications of this package include benchmarking the stability and accuracy of state-of-the-art linear solvers and determining whether singular-to-double-precision matrices are indeed singular. Computationally, this article evaluates the impact of several novel pivoting schemes in exact arithmetic, benchmarks the exact iterative solvers within Linbox, and benchmarks the accuracy of MATLAB sparse backslash. Most importantly, it is shown that SPEX Left LU outperforms the exact iterative solvers in run time on easy instances and in stability as the iterative solver fails on a sizeable subset of the tested (both easy and hard) instances. The SPEX Left LU package is written in ANSI C, comes with a MATLAB interface, and is distributed via GitHub, as a component of the SPEX software package, and as a component of SuiteSparse. 

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5. Sparse autoregressive models for scalable generation of sparse images in particle physics

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