Joint Sparse Coding and Frame Optimization

We present two extensions of the Split Augmented Lagrangian Shrinkage Algorithm (SALSA), each addressing the joint optimization of a parameterized tight frame and their corresponding sparse coefficients. The two extensions, Parameter ADMM and Unrolled SALSA, can be adapted for either Basis Pursuit (constrained) or Basis Pursuit Denoising (unconstrained). The algorithms showcase the flexibility of the recently proposed Enveloped Sinusoid Parseval (ESP) Frames, and in particular, their aptitude for morphological component analysis (a sparsity-based approach to source separation) even when the source sparsity models have unknown parameters. We present a sample application using ESP frames to disentangle an additive mixture of two signal components, and empirically evaluate each algorithm's accuracy in recovering the components' parameters, their robustness in the presence of Gaussian white noise, and compare their performance to a traditional nonlinear least squares approach. The results indicate that Parameter ADMM and Unrolled SALSA outperform nonlinear least squares in low noise settings.