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

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.