arXiv subjects

Alexandre Pasco

Publications and source records attributed to Alexandre Pasco.

4 recordsLinked to original sources

arXiv2026

Surrogate to Poincar\'e inequalities on manifolds for structured dimension reduction in nonlinear feature spaces

This paper is concerned with the approximation of continuously differentiable functions with high-dimensional input by a composition of two functions: a feature map that extracts few features from the input space, and a profile function that approximates the target function taking the features as its low-dimensional input. We focus on the construction of structured nonlinear feature maps, that extract features on separate groups of variables, using a recently introduced gradient-based method that leverages Poincar\'e inequalities on nonlinear manifolds. This method consists in minimizing a non-convex loss functional, which can be a challenging task, especially for small training samples. We first investigate a collective setting, in which we construct a feature map suitable to a parametrized family of high-dimensional functions. In this setting we introduce a new quadratic surrogate to the non-convex loss function and show an upper bound on the latter. We then investigate a grouped setting, in which we construct separate feature maps for separate groups of inputs, and we show that this setting is almost equivalent to multiple collective settings, one for each group of variables.

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arXiv2025

Surrogate to Poincar\'e inequalities on manifolds for dimension reduction in nonlinear feature spaces

We aim to approximate a continuously differentiable function $u:\mathbb{R}^d \rightarrow \mathbb{R}$ by a composition of functions $f\circ g$ where $g:\mathbb{R}^d \rightarrow \mathbb{R}^m$, $m\leq d$, and $f : \mathbb{R}^m \rightarrow \mathbb{R}$ are built in a two stage procedure. For a fixed $g$, we build $f$ using classical regression methods, involving evaluations of $u$. Recent works proposed to build a nonlinear $g$ by minimizing a loss function $\mathcal{J}(g)$ derived from Poincar\'e inequalities on manifolds, involving evaluations of the gradient of $u$. A problem is that minimizing $\mathcal{J}$ may be a challenging task. Hence in this work, we introduce new convex surrogates to $\mathcal{J}$. Leveraging concentration inequalities, we provide suboptimality results for a class of functions $g$, including polynomials, and a wide class of input probability measures. We investigate performances on different benchmarks for various training sample sizes. We show that our approach outperforms standard iterative methods for minimizing the training Poincar\'e inequality based loss, often resulting in better approximation errors, especially for small training sets and $m=1$.

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arXiv2023

Dictionary-based model reduction for state estimation

We consider the problem of state estimation from a few linear measurements, where the state to recover is an element of the manifold $\mathcal{M}$ of solutions of a parameter-dependent equation. The state is estimated using prior knowledge on $\mathcal{M}$ coming from model order reduction. Variational approaches based on linear approximation of $\mathcal{M}$, such as PBDW, yields a recovery error limited by the Kolmogorov width of $\mathcal{M}$. To overcome this issue, piecewise-affine approximations of $\mathcal{M}$ have also been considered, that consist in using a library of linear spaces among which one is selected by minimizing some distance to $\mathcal{M}$. In this paper, we propose a state estimation method relying on dictionary-based model reduction, where a space is selected from a library generated by a dictionary of snapshots, using a distance to the manifold. The selection is performed among a set of candidate spaces obtained from a set of $\ell_1$-regularized least-squares problems. Then, in the framework of parameter-dependent operator equations (or PDEs) with affine parametrizations, we provide an efficient offline-online decomposition based on randomized linear algebra, that ensures efficient and stable computations while preserving theoretical guarantees.

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arXiv2021

Random sketching of operators with application to learning preconditioners

We propose a new random sketching approach for embedding high-dimensional Hilbert-Schmidt operators, using random input-output pairs. Such operator can then be approximated in a low-dimensional subspace of operators by solving a small least-squares problem. To achieve computational efficiency, we introduce a structured random map, composed of three random matrices. We provide rigorous conditions under which subspaces of operators are accurately embedded with high probability. The framework is flexible, as the random matrices may be adapted to the operator structure and the computational environment. As an application, we consider the construction of preconditioners for high-dimensional linear equations. We derive a rigorous characterization of preconditioner quality through the discrepancy between the preconditioned operator and an optimal baseline, which can be tailored to a linear approximation space for the solution. We show that this quantity can be efficiently minimized within the proposed framework, especially for parameter separable linear equations. We then establish rigorous high-probability bounds on the quasi-optimality error of the preconditioned Galerkin projection and on the accuracy of a preconditioned residual-based error estimator when the sketch dimensions are sufficiently large. Numerical experiments on an acoustic wave scattering benchmark demonstrate the effectiveness of the method.

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