Sparse Dynamic Filtering via Earth Mover's Distance Regularization

Tracking time-varying signals is an important task for practical systems working with large discretized domains. Under such settings, sparsity-based approaches improve tracking accuracy since typically few targets appear in the scene (i.e. few locations in the discretized space are occupied). Discretization introduces a unique challenge: the traditional $\ell_{p}$ -norm dynamic constraints produce significant errors when there is even a small spatial mismatch between the predicted and true state. To overcome this, we present a tracking algorithm leveraging concepts from optimal transport, namely utilizing the earth-movers distance (EMD) as a dynamic regularizer to the $\ell_{1}$ -regularized inference problem (i.e., LASSO [1], or BPDN [2]). We extend the problem formulation to complex valued signals and modify the optimization program to reduce the computational burden. We demonstrate the efficacy of our approach in imaging and frequency tracking applications.