Fast Length-Squared Sampling for Positive-Semidefinite Matrices

• URL: http://arxiv.org/abs/2608.12503v1
• Date: Wed, 12 Aug 2026 18:33:22 GMT
• Title: Fast Length-Squared Sampling for Positive-Semidefinite Matrices
• Abstract summary: We describe a simple rejection-sampling-based algorithm to perform length-squared sampling on a matrix.<n>The algorithm runs in just $On)$ expected time, which is significantly sublinear in the input size.
• Score: 15.092552206493544
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We describe a simple rejection-sampling-based algorithm to perform length-squared sampling on an $n \times n$ positive-semidefinite (psd) matrix: that is, to sample a column with probability proportional to its squared $\ell_2$-norm. The algorithm runs in just $O(n)$ expected time, which is significantly sublinear in the input matrix size. The runtime is optimal, even when the input is assumed to be diagonal. Our result has several applications. Length-squared sampling is used by a number of sublinear time algorithms for matrix problems, like low-rank approximation and eigenvalue approximation. Often, it is assumed that the algorithm is given access to the matrix column norms, and thus can perform length-squared sampling efficiently. Our result shows that, at least for psd matrices, we can remove this assumption. We also discuss an application to an asymptotically optimal algorithm for estimating the Frobenius norm of a psd matrix to relative error. Finally, we show that our sampling algorithm yields a very simple sublinear time algorithm for the robust psd low-rank approximation problem introduced by Bakshi et al. (FOCS, 2020), which nearly matches the more complex method developed there.

Related papers

• Improving Algorithmic Efficiency using Cryptography [11.496343300483904]
  We show how to use cryptography to improve the time complexity of solving computational problems.<n>We show that under standard cryptographic assumptions, we can design algorithms that are determinantally faster than existing ones.
  arXiv  Detail & Related papers  (2025-02-18T17:08:59Z)
• Randomized Algorithms for Symmetric Nonnegative Matrix Factorization [1.991322361612643]
  Symmetric Nonnegative Matrix Factorization (SymNMF) is a technique in data analysis and machine learning.<n>We develop two randomized algorithms for its computation.<n>We show that our methods approximately maintain solution quality and achieve significant speed ups for both large dense and large sparse problems.
  arXiv  Detail & Related papers  (2024-02-13T00:02:05Z)
• Fast Minimization of Expected Logarithmic Loss via Stochastic Dual Averaging [8.990961435218544]
  We propose a first-order algorithm named $B$-sample dual averaging with the logarithmic barrier. For the Poisson inverse problem, our algorithm attains an $varepsilon$ solution in $smashtildeO(d3/varepsilon2)$ time. When computing the maximum-likelihood estimate for quantum state tomography, our algorithm yields an $varepsilon$-optimal solution in $smashtildeO(d3/varepsilon2)$ time.
  arXiv  Detail & Related papers  (2023-11-05T03:33:44Z)
• Learning the Positions in CountSketch [49.57951567374372]
  We consider sketching algorithms which first compress data by multiplication with a random sketch matrix, and then apply the sketch to quickly solve an optimization problem. In this work, we propose the first learning-based algorithms that also optimize the locations of the non-zero entries.
  arXiv  Detail & Related papers  (2023-06-11T07:28:35Z)
• Quantum algorithms for spectral sums [50.045011844765185]
  We propose new quantum algorithms for estimating spectral sums of positive semi-definite (PSD) matrices. We show how the algorithms and techniques used in this work can be applied to three problems in spectral graph theory.
  arXiv  Detail & Related papers  (2020-11-12T16:29:45Z)
• Linear-Sample Learning of Low-Rank Distributions [56.59844655107251]
  We show that learning $ktimes k$, rank-$r$, matrices to normalized $L_1$ distance requires $Omega(frackrepsilon2)$ samples. We propose an algorithm that uses $cal O(frackrepsilon2log2fracepsilon)$ samples, a number linear in the high dimension, and nearly linear in the matrices, typically low, rank proofs.
  arXiv  Detail & Related papers  (2020-09-30T19:10:32Z)
• Private Stochastic Convex Optimization: Optimal Rates in Linear Time [74.47681868973598]
  We study the problem of minimizing the population loss given i.i.d. samples from a distribution over convex loss functions. A recent work of Bassily et al. has established the optimal bound on the excess population loss achievable given $n$ samples. We describe two new techniques for deriving convex optimization algorithms both achieving the optimal bound on excess loss and using $O(minn, n2/d)$ gradient computations.
  arXiv  Detail & Related papers  (2020-05-10T19:52:03Z)
• Non-Adaptive Adaptive Sampling on Turnstile Streams [57.619901304728366]
  We give the first relative-error algorithms for column subset selection, subspace approximation, projective clustering, and volume on turnstile streams that use space sublinear in $n$. Our adaptive sampling procedure has a number of applications to various data summarization problems that either improve state-of-the-art or have only been previously studied in the more relaxed row-arrival model.
  arXiv  Detail & Related papers  (2020-04-23T05:00:21Z)
• Optimal Iterative Sketching with the Subsampled Randomized Hadamard Transform [64.90148466525754]
  We study the performance of iterative sketching for least-squares problems. We show that the convergence rate for Haar and randomized Hadamard matrices are identical, andally improve upon random projections. These techniques may be applied to other algorithms that employ randomized dimension reduction.
  arXiv  Detail & Related papers  (2020-02-03T16:17:50Z)

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.