Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.04338

Computing Lewis Weights to High Precision by Fixed-Point Iteration

Swati Padmanabhan
View original source ↗Read source document ↗Export metadata ↓

Abstract

The $\ell_p$-Lewis weights of a matrix are defined by a fixed-point equation. For $p<4$, Cohen and Peng [CP15] showed that iterating an equivalent rearrangement of this equation computes Lewis weights to high precision; for $p\geq4$, prior high-precision methods instead use optimization-based approaches. We show that the direct Lewis fixed-point iteration, appearing in the thesis of Lee [Lee16], computes Lewis weights to high precision for every $p>2$. For a matrix $\mathbf{A}\in\mathbb{R}^{m\times n}$ partitioned into row blocks $\mathbf{A}_{[1]},\ldots,\mathbf{A}_{[k]}$, we compute, for $p>2$, coordinatewise $\varepsilon$-approximate $\ell_p$ block Lewis weights in $O\left(p\log\frac{p\sqrt{\sum_{i=1}^k\operatorname{rank}(\mathbf{A}_{[i]})}}{\varepsilon}\right)$ rounds of exact leverage-score-vector computations. For ordinary Lewis weights, this becomes $O\left(p\log\frac{p\sqrt{m}}{\varepsilon}\right)$, improving the $O\left(p^2\log(m/\varepsilon)\right)$ bound of Gribling, Sidford, and Zhang [GSZ26] for $p\geq4$. Our main observation is that each direct Lewis update contracts the KL divergence to the true weights by a factor of $1-\frac{2}{p}$. We also give an alternate explanation of this contraction through volume sampling and entropic independence. Synthetic experiments closely match our predicted local contraction rates and iteration counts grow approximately linearly with $p$; real-data experiments illustrate the information-concentration tradeoff of finite-$p$ block Lewis designs.

Explore related subjects

cs.DScs.NAmath.NA

Keep this discovery

BibTeXRIS

Swati Padmanabhan. 2026-09-03. Computing Lewis Weights to High Precision by Fixed-Point Iteration. https://arxiv.org/abs/2609.04338

Cite the original work for its findings. Save a collection to share your selection of sources.

Discover connections

cs.DS subjectcs.NA subjectmath.NA subject

Connections use source metadata and explicit phrase matches, not verified experimental comparisons.

KEEP EXPLORING

Related papers

arXiv2026

Data-efficient Kernel Methods for Learning Hamiltonian Systems

Hamiltonian dynamics describe a wide range of physical systems. As such, data-driven simulations of Hamiltonian systems are important for many scientific and engineering problems. In this work, we propose kernel-based methods for identifying and forecasting Hamiltonian systems directly from trajectory data. We present two approaches: a 2-step method that reconstructs trajectories before learning the Hamiltonian, and a 1-step method that jointly infers both. Across several benchmark systems, including mass-spring dynamics, a nonlinear pendulum, and the Henon-Heiles system, we demonstrate that our framework achieves accurate, data-efficient predictions and outperforms 2-step kernel-based baselines, particularly in scarce-data regimes, while preserving the Hamiltonian structure. Moreover, we prove a priori error estimates, ensuring reliability of the learned models. We also provide a more general, problem-agnostic numerical framework that goes beyond Hamiltonian systems and can be used for data-driven learning of arbitrary dynamical systems.

math.NA↗
arXiv2026

Smoothed Picard Hamiltonian Monte Carlo

We develop a new low-accuracy sampler, called \emph{smoothed Picard Hamiltonian Monte Carlo}, which combines Gaussian smoothing, Picard iteration, and higher-order discretization. For a log-concave target $π\propto \exp(-V)$ in dimension $d$ satisfying $0 \prec αI \preceq \nabla^2 V \preceq βI$, with condition number $κ:= β/α$, smoothed Picard HMC returns a sample with $\sqrt α\,W_2(\cdot,π) \le \varepsilon$ using $\widetilde O(κ^2 + κ^{7/6} d^{1/6}/\varepsilon^{1/3})$ gradient queries. We also prove stronger $W_q$ bounds, and then develop an algorithmic framework, the recursive warm start generator, to upgrade these $W_q$ bounds to stronger divergence guarantees. This produces a warm start for the proximal bouncy particle sampler, introduced in a companion work, leading to a high-accuracy log-concave sampler with complexity $\widetilde O((κ^{7/6} d^{1/6} + κ^{1/2} d^{1/4})\mathrm{polylog}(1/\varepsilon))$.

math.ST↗
arXiv2025

Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification

The article introduces a method to learn dynamical systems that are governed by Euler--Lagrange equations from data. The method is based on Gaussian process regression and identifies continuous or discrete Lagrangians and is, therefore, structure preserving by design. A rigorous proof of convergence as the distance between observation data points converges to zero and lower bounds for convergence rates are provided. Next to convergence guarantees, the method allows for quantification of model uncertainty, which can provide a basis of adaptive sampling techniques. We provide efficient uncertainty quantification of any observable that is linear in the Lagrangian, including of Hamiltonian functions (energy) and symplectic structures, which is of interest in the context of system identification. The article overcomes major practical and theoretical difficulties related to the ill-posedness of the identification task of (discrete) Lagrangians through a careful design of geometric regularisation strategies and through an exploit of a relation to convex minimisation problems in reproducing kernel Hilbert spaces.

math.NA↗