arXiv subjects

Marc Aurèle Gilles

Publications and source records attributed to Marc Aurèle Gilles.

8 recordsLinked to original sources

arXiv2026

Convergence rates of randomly pivoted methods for low-rank approximation

Randomly pivoted Cholesky, QR, and LU are iterative algorithms that form structured low-rank approximations of a matrix by sampling columns, or rows and columns, from the residual. We give convergence rates for these methods that depend on the decay of the singular values of the original matrix. The rates hold, up to a constant, when pivots are drawn from an approximation to the exact distribution. This extra degree of freedom can be used to trade exact sampling for cheaper approximations. We use it to establish rates for variants of randomly pivoted LU, including one that picks pivots using a sketch of the input matrix.

math.NA↗

arXiv2026

Convergence rates for pivoted QR and LU

Pivoted QR and pivoted LU decompositions are greedy algorithms used to compute low-rank approximations of matrices from selected columns, or selected rows and columns. Despite their practical robustness, general worst-case bounds comparing their errors with those of the best corresponding low-rank approximations contain exponentially growing factors and do not explain their behavior under modest singular value decay. We prove that under approximate greedy pivoting, their error is controlled by the determinant of a submatrix, which is bounded by the geometric mean of the leading singular values. Using this bound, we establish convergence rates under algebraic and geometric singular value decay. We also extend the LU analysis to functions of two variables. By bounding the determinants of arbitrary sampled submatrices, we obtain algebraic convergence rates under differentiability assumptions and geometric convergence under analyticity.

math.NA↗

arXiv2026

Low-Rank Approximation by Randomly Pivoted LU

The low-rank approximation properties of Randomly Pivoted LU (RPLU), a variant of Gaussian elimination where pivots are sampled proportional to the squared entries of the Schur complement, are analyzed. It is shown that the RPLU iterates converge geometrically in expectation for matrices with rapidly decaying singular values. RPLU outperforms existing low-rank approximation algorithms in two settings: first, when memory is limited, RPLU can be implemented with $\mathcal{O}(k^2 + m + n)$ storage and $\mathcal{O}( k(m + n)+ k\mathcal{M}(\mat{A}) + k^3)$ operations, where $\mathcal{M}(\mat{A})$ is the cost of a matvec with $\mat{A}\in\mathbb{C}^{n\times m}$ or its adjoint, for a rank-$k$ approximation. Second, when the matrix and its Schur complements share exploitable structure, such as for Cauchy-like matrices. The efficacy of RPLU is illustrated with several examples, including applications in rational approximation and solving large linear systems on GPUs.

math.NA↗

arXiv2024

Moment-based metrics for molecules computable from cryo-EM images

Single particle cryogenic electron microscopy (cryo-EM) is an imaging technique capable of recovering the high-resolution 3-D structure of biological macromolecules from many noisy and randomly oriented projection images. One notable approach to 3-D reconstruction, known as Kam's method, relies on the moments of the 2-D images. Inspired by Kam's method, we introduce a rotationally invariant metric between two molecular structures, which does not require 3-D alignment. Further, we introduce a metric between a stack of projection images and a molecular structure, which is invariant to rotations and reflections and does not require performing 3-D reconstruction. Additionally, the latter metric does not assume a uniform distribution of viewing angles. We demonstrate uses of the new metrics on synthetic and experimental datasets, highlighting their ability to measure structural similarity.

q-bio.BM↗

arXiv2022

A Molecular Prior Distribution for Bayesian Inference Based on Wilson Statistics

Background and Objective: Wilson statistics describe well the power spectrum of proteins at high frequencies. Therefore, it has found several applications in structural biology, e.g., it is the basis for sharpening steps used in cryogenic electron microscopy (cryo-EM). A recent paper gave the first rigorous proof of Wilson statistics based on a formalism of Wilson's original argument. This new analysis also leads to statistical estimates of the scattering potential of proteins that reveal a correlation between neighboring Fourier coefficients. Here we exploit these estimates to craft a novel prior that can be used for Bayesian inference of molecular structures. Methods: We describe the properties of the prior and the computation of its hyperparameters. We then evaluate the prior on two synthetic linear inverse problems, and compare against a popular prior in cryo-EM reconstruction at a range of SNRs. Results: We show that the new prior effectively suppresses noise and fills-in low SNR regions in the spectral domain. Furthermore, it improves the resolution of estimates on the problems considered for a wide range of SNR and produces Fourier Shell Correlation curves that are insensitive to masking effects. Conclusions: We analyze the assumptions in the model, discuss relations to other regularization strategies, and postulate on potential implications for structure determination in cryo-EM.

q-bio.QM↗

arXiv2019

Evasive path planning under surveillance uncertainty

The classical setting of optimal control theory assumes full knowledge of the process dynamics and the costs associated with every control strategy. The problem becomes much harder if the controller only knows a finite set of possible running cost functions, but has no way of checking which of these running costs is actually in place. In this paper we address this challenge for a class of evasive path planning problems on a continuous domain, in which an Evader needs to reach a target while minimizing his exposure to an enemy Observer, who is in turn selecting from a finite set of known surveillance plans. Our key assumption is that both the evader and the observer need to commit to their (possibly probabilistic) strategies in advance and cannot immediately change their actions based on any newly discovered information about the opponent's current position. We consider two types of evader behavior: in the first one, a completely risk-averse evader seeks a trajectory minimizing his {\em worst-case} cumulative observability, and in the second, the evader is concerned with minimizing the {\em average-case} cumulative observability. The latter version is naturally interpreted as a semi-infinite strategic game, and we provide an efficient method for approximating its Nash equilibrium. The proposed approach draws on methods from game theory, convex optimization, optimal control, and multiobjective dynamic programming. We illustrate our algorithm using numerical examples and discuss the computational complexity, including for the generalized version with multiple evaders.

math.OC↗

arXiv2019

A subspace pursuit method to infer refractivity in the marine atmospheric boundary layer

Inferring electromagnetic propagation characteristics within the marine atmospheric boundary layer (MABL) from data in real time is crucial for modern maritime navigation and communications. The propagation of electromagnetic waves is well modeled by a partial differential equation (PDE): a Helmholtz equation. A natural way to solve the MABL characterization inverse problem is to minimize what is observed and what is predicted by the PDE. However, this optimization is difficult because it has many local minima. We propose an alternative solution that relies on the properties of the PDE but does not involve solving the full forward model. Ducted environments result in an EM field which can be decomposed into a few propagating, trapped modes. These modes are a subset of the solutions to a Sturm-Liouville eigenvalue problem. We design a new objective function that measures the distance from the observations to a subspace spanned by these eigenvectors. The resulting optimization problem is much easier than the one that arises in the standard approach, and we show how to solve the associated nonlinear eigenvalue problem efficiently, leading to a real-time method.

physics.ao-ph↗

arXiv2018

Continuous analogues of Krylov methods for differential operators

Analogues of the conjugate gradient method, MINRES, and GMRES are derived for solving boundary value problems (BVPs) involving second-order differential operators. Two challenges arise: imposing the boundary conditions on the solution while building up a Krylov subspace, and guaranteeing convergence of the Krylov-based method on unbounded operators. Our approach employs projection operators to guarantee that the boundary conditions are satisfied, and we develop an operator preconditioner that ensures that an approximate solution is computed after a finite number of iterations. The developed Krylov methods are practical iterative BVP solvers that are particularly efficient when a fast operator-function product is available.

math.NA↗