Title: On statistical estimation, signal denoising, and convex optimization
Abstract: We revisit the classical statistical problem of adaptive discrete-time signal denoising. Conventional nonparametric statistics and signal processing approaches rely on strong structural assumptions on the signal set. The approach we present, inspired by the seminal works from Ibragimov-Khasminskii and later Donoho, takes a different view and focuses instead on statistical estimators amenable to convex optimization. We show that such estimators possess better statistical properties than conventional ones. In particular, under an assumption of approximate shift-invariance, the proposed estimators enjoy l2-loss oracle inequalities. We show how to implement them using optimal first-order methods and highlight interesting statistical-computational trade-offs. Joint work with D. Ostrovskii, A. Juditsky, A. Nemirovski.