Dynamical Programming for off-the-grid dynamic Inverse Problems

TL;DR

This paper introduces new algorithms for reconstructing time-varying data as discrete trajectories using off-the-grid sparse-spikes decomposition, leveraging optimal transport and shortest path algorithms for efficient and globally optimal solutions.

Contribution

It generalizes existing convex variational models for off-the-grid sparse decomposition and develops faster numerical methods with proven convergence to global optima.

Findings

Methods converge to globally optimal reconstructions.

Achieved 100x speedup over previous methods.

Extended framework to unbalanced optimal transport.

Abstract

In this work we consider algorithms for reconstructing time-varying data into a finite sum of discrete trajectories, alternatively, an off-the-grid sparse-spikes decomposition which is continuous in time. Recent work showed that this decomposition was possible by minimising a convex variational model which combined a quadratic data fidelity with dynamical Optimal Transport. We generalise this framework and propose new numerical methods which leverage efficient classical algorithms for computing shortest paths on directed acyclic graphs. Our theoretical analysis confirms that these methods converge to globally optimal reconstructions which represent a finite number of discrete trajectories. Numerically, we show new examples for unbalanced Optimal Transport penalties, and for balanced examples we are 100 times faster in comparison to the previously known method.