Dimension reduction, exact recovery, and error estimates for sparse reconstruction in phase space

TL;DR

This paper introduces a dimension reduction method for phase space reconstruction in inverse problems, enabling exact recovery and error estimation in lower dimensions, thus overcoming computational challenges while maintaining accuracy.

Contribution

A novel dimension reduction technique for phase space reconstruction that preserves exact recovery results and provides new error estimates in noisy settings.

Findings

Proves that exact reconstruction results hold after dimension reduction.

Provides error estimates for noisy data in optimal transport metrics.

Demonstrates the method's effectiveness in superresolution problems.

Abstract

An important theme in modern inverse problems is the reconstruction of time-dependent data from only finitely many measurements. To obtain satisfactory reconstruction results in this setting it is essential to strongly exploit temporal consistency between the different measurement times. The strongest consistency can be achieved by reconstructing data directly in phase space, the space of positions and velocities. However, this space is usually too high-dimensional for feasible computations. We introduce a novel dimension reduction technique, based on projections of phase space onto lower-dimensional subspaces, which provably circumvents this curse of dimensionality: Indeed, in the exemplary framework of superresolution we prove that known exact reconstruction results stay true after dimension reduction, and we additionally prove new error estimates of reconstructions from noisy data in optimal transport metrics which are of the same quality as one would obtain in the non-dimension-reduced case.