# Reconstructing high-dimensional Hilbert-valued functions via compressed sensing

## TL;DR

This paper introduces a new sparse polynomial method for approximating high-dimensional Hilbert-valued functions, with theoretical guarantees and an algorithm, applied to parameterized PDEs with stochastic inputs.

## Contribution

It develops a mixed-norm $$ regularization technique for joint sparse recovery of Hilbert-valued functions, extending compressed sensing theory to infinite-dimensional settings.

## Key findings

- Proves recovery guarantees for Hilbert-valued functions.
- Provides an algorithm with strong convergence in infinite dimensions.
- Demonstrates minimal sample complexity through numerical experiments.

## Abstract

We present and analyze a novel sparse polynomial technique for approximating high-dimensional Hilbert-valued functions, with application to parameterized partial differential equations (PDEs) with deterministic and stochastic inputs. Our theoretical framework treats the function approximation problem as a joint sparse recovery problem, where the set of jointly sparse vectors is possibly infinite. To achieve the simultaneous reconstruction of Hilbert-valued functions in both parametric domain and Hilbert space, we propose a novel mixed-norm based $\ell_1$ regularization method that exploits both energy and sparsity. Our approach requires extensions of concepts such as the restricted isometry and null space properties, allowing us to prove recovery guarantees for sparse Hilbert-valued function reconstruction. We complement the enclosed theory with an algorithm for Hilbert-valued recovery, based on standard forward-backward algorithm, meanwhile establishing its strong convergence in the considered infinite-dimensional setting. Finally, we demonstrate the minimal sample complexity requirements of our approach, relative to other popular methods, with numerical experiments approximating the solutions of high-dimensional parameterized elliptic PDEs.