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1. Near-optimal hierarchical matrix approximation from matrix-vector products

   Chen, Tyler; Keles, Feyza Duman; Halikias, Diana; Musco, Cameron; Musco, Christopher; Persson, David (January 2025, ACM-SIAM Symposium on Discrete Algorithms)

   We describe a randomized algorithm for producing a near-optimal hierarchical off-diagonal low-rank (HODLR) approximation to an n × n matrix A, accessible only though matrix-vector products with A and AT. We prove that, for the rank-k HODLR approximation problem, our method achieves a (1 + β )log(n )-optimal approximation in expected Frobenius norm using O (k log(n )/β3) matrix-vector products. In particular, the algorithm obtains a (1 + ∈ )-optimal approximation with O (k log4(n )/∈3) matrix-vector products, and for any constant c, an nc-optimal approximation with O (k log(n )) matrix-vector products. Apart from matrix-vector products, the additional computational cost of our method is just O (n poly(log(n ), k, β )). We complement the upper bound with a lower bound, which shows that any matrix-vector query algorithm requires at least Ω(k log(n ) + k/ε ) queries to obtain a (1 + ε )-optimal approximation. Our algorithm can be viewed as a robust version of widely used “peeling” methods for recovering HODLR matrices and is, to the best of our knowledge, the first matrix-vector query algorithm to enjoy theoretical worst- case guarantees for approximation by any hierarchical matrix class. To control the propagation of error between levels of hierarchical approximation, we introduce a new perturbation bound for low-rank approximation, which shows that the widely used Generalized Nyström method enjoys inherent stability when implemented with noisy matrix-vector products. We also introduce a novel randomly perforated matrix sketching method to further control the error in the peeling algorithm. 

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2. Querying a Matrix through Matrix-Vector Products

   https://doi.org/10.1145/3470566  

   Sun, Xiaoming; Woodruff, David P.; Yang, Guang; Zhang, Jialin (October 2021, ACM Transactions on Algorithms)

   We consider algorithms with access to an unknown matrix M ε F n×d via matrix-vector products , namely, the algorithm chooses vectors v 1 , ⃛ , v q , and observes Mv 1 , ⃛ , Mv q . Here the v i can be randomized as well as chosen adaptively as a function of Mv 1 , ⃛ , Mv i-1 . Motivated by applications of sketching in distributed computation, linear algebra, and streaming models, as well as connections to areas such as communication complexity and property testing, we initiate the study of the number q of queries needed to solve various fundamental problems. We study problems in three broad categories, including linear algebra, statistics problems, and graph problems. For example, we consider the number of queries required to approximate the rank, trace, maximum eigenvalue, and norms of a matrix M; to compute the AND/OR/Parity of each column or row of M, to decide whether there are identical columns or rows in M or whether M is symmetric, diagonal, or unitary; or to compute whether a graph defined by M is connected or triangle-free. We also show separations for algorithms that are allowed to obtain matrix-vector products only by querying vectors on the right, versus algorithms that can query vectors on both the left and the right. We also show separations depending on the underlying field the matrix-vector product occurs in. For graph problems, we show separations depending on the form of the matrix (bipartite adjacency versus signed edge-vertex incidence matrix) to represent the graph. Surprisingly, very few works discuss this fundamental model, and we believe a thorough investigation of problems in this model would be beneficial to a number of different application areas. 

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3. Structured FISTA for image restoration

   https://doi.org/10.1002/nla.2278  

   Chen, Zixuan; Nagy, James G.; Xi, Yuanzhe; Yu, Bo (December 2019, Numerical Linear Algebra with Applications)

   Summary In this paper, we propose an efficient numerical scheme for solving some large‐scale ill‐posed linear inverse problems arising from image restoration. In order to accelerate the computation, two different hidden structures are exploited. First, the coefficient matrix is approximated as the sum of a small number of Kronecker products. This procedure not only introduces one more level of parallelism into the computation but also enables the usage of computationally intensive matrix–matrix multiplications in the subsequent optimization procedure. We then derive the corresponding Tikhonov regularized minimization model and extend the fast iterative shrinkage‐thresholding algorithm (FISTA) to solve the resulting optimization problem. Because the matrices appearing in the Kronecker product approximation are all structured matrices (Toeplitz, Hankel, etc.), we can further exploit their fast matrix–vector multiplication algorithms at each iteration. The proposed algorithm is thus calledstructuredFISTA (sFISTA). In particular, we show that the approximation error introduced by sFISTA is well under control and sFISTA can reach the same image restoration accuracy level as FISTA. Finally, both the theoretical complexity analysis and some numerical results are provided to demonstrate the efficiency of sFISTA. 

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4. Robust PCA by Manifold Optimization

   Zhang, Teng; Yang, Yi (November 2018, Journal of machine learning research)

   Robust PCA is a widely used statistical procedure to recover an underlying low-rank matrix with grossly corrupted observations. This work considers the problem of robust PCA as a nonconvex optimization problem on the manifold of low-rank matrices and proposes two algorithms based on manifold optimization. It is shown that, with a properly designed initialization, the proposed algorithms are guaranteed to converge to the underlying lowrank matrix linearly. Compared with a previous work based on the factorization of low-rank matrices Yi et al. (2016), the proposed algorithms reduce the dependence on the condition number of the underlying low-rank matrix theoretically. Simulations and real data examples con rm the competitive performance of our method. 

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5. Randomized Algorithms for Symmetric Nonnegative Matrix Factorization

   https://doi.org/10.1137/24M1638355  

   Hayashi, Koby; Aksoy, Sinan G; Ballard, Grey; Park, Haesun (March 2025, SIAM Journal on Matrix Analysis and Applications)

   Symmetric Nonnegative Matrix Factorization (SymNMF) is a technique in data analysis and machine learning that approximates a symmetric matrix with a product of a nonnegative, low-rank matrix and its transpose. To design faster and more scalable algorithms for SymNMF, we develop two randomized algorithms for its computation. The first algorithm uses randomized matrix sketching to compute an initial low-rank approximation to the input matrix and proceeds to rapidly compute a SymNMF of the approximation. The second algorithm uses randomized leverage score sampling to approximately solve constrained least squares problems. Many successful methods for SymNMF rely on (approximately) solving sequences of constrained least squares problems. We prove theoretically that leverage score sampling can approximately solve nonnegative least squares problems to a chosen accuracy with high probability. Additionally, we prove sampling complexity results for previously proposed hybrid sampling techniques which deterministically include high leverage score rows. This hybrid scheme is crucial for obtaining speedups in practice. Finally, we demonstrate that both methods work well in practice by applying them to graph clustering tasks on large real world data sets. These experiments show that our methods approximately maintain solution quality and achieve significant speedups for both large dense and large sparse problems. 

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