Near-Optimal Algorithms for Linear Algebra in the Current Matrix Multiplication Time

• URL: http://arxiv.org/abs/2107.08090v1
• Date: Fri, 16 Jul 2021 19:34:10 GMT
• Title: Near-Optimal Algorithms for Linear Algebra in the Current Matrix Multiplication Time
• Authors: Nadiia Chepurko, Kenneth L. Clarkson, Praneeth Kacham and David P. Woodruff
• Abstract summary: We show how to bypass the main open question of Nelson and Nguyen (FOCS, 2013) regarding the logarithmic factors in the sketching dimension for existing constant factor approximation oblivious subspace embeddings. A key technique we use is an explicit mapping of Indyk based on uncertainty principles and extractors. For the fundamental problems of rank computation and finding a linearly independent subset of columns, our algorithms improve Cheung, Kwok, and Lau (JACM, 2013) and are optimal to within a constant factor and a $loglog(n)$-factor, respectively.
• Score: 46.31710224483631
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: Currently, in the numerical linear algebra community, it is thought that to obtain nearly-optimal bounds for various problems such as rank computation, finding a maximal linearly independent subset of columns, regression, low rank approximation, maximum matching on general graphs and linear matroid union, one would need to resolve the main open question of Nelson and Nguyen (FOCS, 2013) regarding the logarithmic factors in the sketching dimension for existing constant factor approximation oblivious subspace embeddings. We show how to bypass this question using a refined sketching technique, and obtain optimal or nearly optimal bounds for these problems. A key technique we use is an explicit mapping of Indyk based on uncertainty principles and extractors, which after first applying known oblivious subspace embeddings, allows us to quickly spread out the mass of the vector so that sampling is now effective, and we avoid a logarithmic factor that is standard in the sketching dimension resulting from matrix Chernoff bounds. For the fundamental problems of rank computation and finding a linearly independent subset of columns, our algorithms improve Cheung, Kwok, and Lau (JACM, 2013) and are optimal to within a constant factor and a $\log\log(n)$-factor, respectively. Further, for constant factor regression and low rank approximation we give the first optimal algorithms, for the current matrix multiplication exponent.

Related papers

• Riemannian Optimization for Non-convex Euclidean Distance Geometry with Global Recovery Guarantees [6.422262171968397]
  Two algorithms are proposed to solve the Euclidean Distance Geometry problem. First algorithm converges linearly to the true solution. Second algorithm demonstrates strong numerical performance on both synthetic and real data.
  arXiv  Detail & Related papers  (2024-10-08T21:19:22Z)
• An Efficient Algorithm for Clustered Multi-Task Compressive Sensing [60.70532293880842]
  Clustered multi-task compressive sensing is a hierarchical model that solves multiple compressive sensing tasks. The existing inference algorithm for this model is computationally expensive and does not scale well in high dimensions. We propose a new algorithm that substantially accelerates model inference by avoiding the need to explicitly compute these covariance matrices.
  arXiv  Detail & Related papers  (2023-09-30T15:57:14Z)
• Refined Regret for Adversarial MDPs with Linear Function Approximation [50.00022394876222]
  We consider learning in an adversarial Decision Process (MDP) where the loss functions can change arbitrarily over $K$ episodes. This paper provides two algorithms that improve the regret to $tildemathcal O(K2/3)$ in the same setting.
  arXiv  Detail & Related papers  (2023-01-30T14:37:21Z)
• Alternating Mahalanobis Distance Minimization for Stable and Accurate CP Decomposition [4.847980206213335]
  We introduce a new formulation for deriving singular values and vectors of a tensor by considering the critical points of a function different from what is used in the previous work. We show that a subsweep of this algorithm can achieve a superlinear convergence rate for exact CPD with known rank. We then view the algorithm as optimizing a Mahalanobis distance with respect to each factor with ground metric dependent on the other factors.
  arXiv  Detail & Related papers  (2022-04-14T19:56:36Z)
• High-Dimensional Sparse Bayesian Learning without Covariance Matrices [66.60078365202867]
  We introduce a new inference scheme that avoids explicit construction of the covariance matrix. Our approach couples a little-known diagonal estimation result from numerical linear algebra with the conjugate gradient algorithm. On several simulations, our method scales better than existing approaches in computation time and memory.
  arXiv  Detail & Related papers  (2022-02-25T16:35:26Z)
• Nearly Optimal Linear Convergence of Stochastic Primal-Dual Methods for Linear Programming [5.126924253766052]
  We show that the proposed method exhibits a linear convergence rate for solving sharp instances with a high probability. We also propose an efficient coordinate-based oracle for unconstrained bilinear problems.
  arXiv  Detail & Related papers  (2021-11-10T04:56:38Z)
• A spectral algorithm for robust regression with subgaussian rates [0.0]
  We study a new linear up to quadratic time algorithm for linear regression in the absence of strong assumptions on the underlying distributions of samples. The goal is to design a procedure which attains the optimal sub-gaussian error bound even though the data have only finite moments.
  arXiv  Detail & Related papers  (2020-07-12T19:33:50Z)
• Effective Dimension Adaptive Sketching Methods for Faster Regularized Least-Squares Optimization [56.05635751529922]
  We propose a new randomized algorithm for solving L2-regularized least-squares problems based on sketching. We consider two of the most popular random embeddings, namely, Gaussian embeddings and the Subsampled Randomized Hadamard Transform (SRHT)
  arXiv  Detail & Related papers  (2020-06-10T15:00:09Z)
• Optimal Randomized First-Order Methods for Least-Squares Problems [56.05635751529922]
  This class of algorithms encompasses several randomized methods among the fastest solvers for least-squares problems. We focus on two classical embeddings, namely, Gaussian projections and subsampled Hadamard transforms. Our resulting algorithm yields the best complexity known for solving least-squares problems with no condition number dependence.
  arXiv  Detail & Related papers  (2020-02-21T17:45:32Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.