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

TL;DR

The paper addresses reconstructing moving particle configurations from finitely many measurements by embedding the problem into phase space and applying a novel dimension-reduction through Radon-type projections. It shows that exact recovery and stability results from the high-dimensional lifted formulation persist under dimension reduction, and it develops a robust duality-based framework to derive error bounds using unbalanced transport and Bregman distances. The contributions include (i) a detailed dimension-reduced convex program, (ii) exact-recovery results under dynamic regularity and ghost/coincidence-free projections, (iii) convergence rates for noisy data via unbalanced Wasserstein metrics, and (iv) a discretization pipeline with numerical validation against full-dimensional methods on truncated Fourier measurements. These results enable accurate, scalable reconstruction of moving particles from limited temporal measurements, with broad applicability to dynamic inverse problems in imaging and sensing.

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.

Figures (3)

• Figure 1: Example grids and projections. The green line in projection direction $\theta$ is subdivided into red bins corresponding to the grid $G_r(\theta,t)$. The contribution of a grid cell to the projection onto a red bin $[r_i, r_{i+1}]$ is determined by its relative area of intersection (visualized by opacity) with the strip orthogonal to $\theta$.
• Figure 2: Top row: Comparing static reconstruction to our dynamic method with parameters as in dimred. mid, applied to the $D_{3}$ dataset with 3, 5 and 7 measurement times (left to right). Results on each configuration evaluated by the clustering method as "correct" or "failed" and divided into four groups per number of measurement times. Bottom row: Comparing the correct reconstruction rate of our dimension-reduced methods against static reconstruction and ADCG applied to the full-dimensional model. Datasets used are $D_{3}$, $D_{5}$, $D_{7}$ (left to right), particle configurations are binned by their dynamic separation, and the rate of correctly reconstructed configurations, as evaluated by the clustering method, is plotted for each bin.
• Figure 3: Convergence properties for vanishing noise level for our method applied to datasets with one, two and $4-20$ particles in each configuration, as well as ADCG on the latter. Slope $\sqrt{\delta}$ provided for comparison.

Theorems & Definitions (134)

• Definition 1: Coincidence and ghost particle
• Theorem 3: Exact reconstruction for noise-free data -- Hausdorff measure
• Theorem 4: Exact reconstruction for noise-free data -- Counting measure
• Definition 5: Stably reconstructible measure
• Theorem 6: Reconstruction error estimates
• Definition 7: Borel and Radon measure, total variation
• Theorem 8: Duality, rudin2006real_complex_analysis_mh
• Definition 9: Support of a measure
• Proposition 10: Support properties
• proof