Approaching Optimality for Solving Dense Linear Systems with Low-Rank Structure

• URL: http://arxiv.org/abs/2507.11724v1
• Date: Tue, 15 Jul 2025 20:48:30 GMT
• Title: Approaching Optimality for Solving Dense Linear Systems with Low-Rank Structure
• Authors: Michał Dereziński, Aaron Sidford,
• Abstract summary: We provide new high-accuracy randomized algorithms for solving linear systems and regression problems.<n>Our algorithms nearly-match a natural complexity limit under dense inputs for these problems.<n>We show how to obtain these running times even under the weaker assumption that all but $k$ of the singular values have a bounded generalized mean.
• Score: 16.324043075920564
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We provide new high-accuracy randomized algorithms for solving linear systems and regression problems that are well-conditioned except for $k$ large singular values. For solving such $d \times d$ positive definite system our algorithms succeed whp. and run in time $\tilde O(d^2 + k^\omega)$. For solving such regression problems in a matrix $\mathbf{A} \in \mathbb{R}^{n \times d}$ our methods succeed whp. and run in time $\tilde O(\mathrm{nnz}(\mathbf{A}) + d^2 + k^\omega)$ where $\omega$ is the matrix multiplication exponent and $\mathrm{nnz}(\mathbf{A})$ is the number of non-zeros in $\mathbf{A}$. Our methods nearly-match a natural complexity limit under dense inputs for these problems and improve upon a trade-off in prior approaches that obtain running times of either $\tilde O(d^{2.065}+k^\omega)$ or $\tilde O(d^2 + dk^{\omega-1})$ for $d\times d$ systems. Moreover, we show how to obtain these running times even under the weaker assumption that all but $k$ of the singular values have a suitably bounded generalized mean. Consequently, we give the first nearly-linear time algorithm for computing a multiplicative approximation to the nuclear norm of an arbitrary dense matrix. Our algorithms are built on three general recursive preconditioning frameworks, where matrix sketching and low-rank update formulas are carefully tailored to the problems' structure.

Related papers

• Quantum Algorithms for Projection-Free Sparse Convex Optimization [32.34794896079469]
  For the vector domain, we propose two quantum algorithms for sparse constraints that find a $varepsilon$-optimal solution with the query complexity of $O(sqrtd/varepsilon)$.<n>For the matrix domain, we propose two quantum algorithms for nuclear norm constraints that improve the time complexity to $tildeO(rd/varepsilon2)$ and $tildeO(sqrtrd/varepsilon3)$.
  arXiv  Detail & Related papers  (2025-07-11T12:43:58Z)
• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• Optimal Sketching for Residual Error Estimation for Matrix and Vector Norms [50.15964512954274]
  We study the problem of residual error estimation for matrix and vector norms using a linear sketch. We demonstrate that this gives a substantial advantage empirically, for roughly the same sketch size and accuracy as in previous work. We also show an $Omega(k2/pn1-2/p)$ lower bound for the sparse recovery problem, which is tight up to a $mathrmpoly(log n)$ factor.
  arXiv  Detail & Related papers  (2024-08-16T02:33:07Z)
• Faster Linear Systems and Matrix Norm Approximation via Multi-level Sketched Preconditioning [10.690769339903941]
  We present a new class of preconditioned iterative methods for solving linear systems of the form $Ax = b$.<n>Our methods are based on constructing a low-rank Nystr"om approximation to $A$ using sparse random sketching matrix.<n>We prove that the convergence of our methods depends on a natural average condition number of $A$, which improves as the rank of the Nystr"om approximation increases.
  arXiv  Detail & Related papers  (2024-05-09T15:53:43Z)
• Solving Dense Linear Systems Faster Than via Preconditioning [1.8854491183340518]
  We show that our algorithm has an $tilde O(n2)$ when $k=O(n0.729)$. In particular, our algorithm has an $tilde O(n2)$ when $k=O(n0.729)$. Our main algorithm can be viewed as a randomized block coordinate descent method.
  arXiv  Detail & Related papers  (2023-12-14T12:53:34Z)
• Structured Semidefinite Programming for Recovering Structured Preconditioners [41.28701750733703]
  We give an algorithm which, given positive definite $mathbfK in mathbbRd times d$ with $mathrmnnz(mathbfK)$ nonzero entries, computes an $epsilon$-optimal diagonal preconditioner in time. We attain our results via new algorithms for a class of semidefinite programs we call matrix-dictionary approximation SDPs.
  arXiv  Detail & Related papers  (2023-10-27T16:54:29Z)
• Efficiently Learning One-Hidden-Layer ReLU Networks via Schur Polynomials [50.90125395570797]
  We study the problem of PAC learning a linear combination of $k$ ReLU activations under the standard Gaussian distribution on $mathbbRd$ with respect to the square loss. Our main result is an efficient algorithm for this learning task with sample and computational complexity $(dk/epsilon)O(k)$, whereepsilon>0$ is the target accuracy.
  arXiv  Detail & Related papers  (2023-07-24T14:37:22Z)
• Fast $(1+\varepsilon)$-Approximation Algorithms for Binary Matrix Factorization [54.29685789885059]
  We introduce efficient $(1+varepsilon)$-approximation algorithms for the binary matrix factorization (BMF) problem. The goal is to approximate $mathbfA$ as a product of low-rank factors. Our techniques generalize to other common variants of the BMF problem.
  arXiv  Detail & Related papers  (2023-06-02T18:55:27Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• Sketching Algorithms and Lower Bounds for Ridge Regression [65.0720777731368]
  We give a sketching-based iterative algorithm that computes $1+varepsilon$ approximate solutions for the ridge regression problem. We also show that this algorithm can be used to give faster algorithms for kernel ridge regression.
  arXiv  Detail & Related papers  (2022-04-13T22:18:47Z)
• Training (Overparametrized) Neural Networks in Near-Linear Time [21.616949485102342]
  We show how to speed up the algorithm of [CGH+1] for training (mildly overetrized) ReparamLU networks. The centerpiece of our algorithm is to reformulate the Gauss-Newton as an $ell$-recondition.
  arXiv  Detail & Related papers  (2020-06-20T20:26:14Z)
• Maximizing Determinants under Matroid Constraints [69.25768526213689]
  We study the problem of finding a basis $S$ of $M$ such that $det(sum_i in Sv_i v_i v_itop)$ is maximized. This problem appears in a diverse set of areas such as experimental design, fair allocation of goods, network design, and machine learning.
  arXiv  Detail & Related papers  (2020-04-16T19:16:38Z)

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.