Epigraphical Relaxation for Minimizing Layered Mixed Norms

TL;DR

This paper introduces an epigraphical relaxation technique to optimize layered mixed norms that are otherwise non-proximable, enabling efficient regularization in signal processing tasks.

Contribution

The paper presents a novel ERx method that decouples non-proximable mixed norms into manageable components, allowing for broader application in regularization problems.

Findings

ERx effectively handles a wide range of non-proximable mixed norms.

New regularizers based on ERx improve image restoration and low-rank recovery.

Experiments demonstrate the utility and effectiveness of ERx in practical scenarios.

Abstract

This paper proposes an epigraphical relaxation (ERx) technique for non-proximable mixed norm minimization. Mixed norm regularization methods play a central role in signal reconstruction and processing, where their optimization relies on the fact that the proximity operators of the mixed norms can be computed efficiently. To bring out the power of regularization, sophisticated layered modeling of mixed norms that can capture inherent signal structure is a key ingredient, but the proximity operator of such a mixed norm is often unavailable (non-proximable). Our ERx decouples a layered non-proximable mixed norm into a norm and multiple epigraphical constraints. This enables us to handle a wide range of non-proximable mixed norms in optimization, as long as both the proximal operator of the outermost norm and the projection onto each epigraphical constraint are efficiently computable. Moreover, under mild conditions, we prove that ERx does not change the minimizer of the original problem despite relaxing equality constraints into inequality ones. We also develop new regularizers based on ERx: one is decorrelated structure-tensor total variation for color image restoration, and the other is amplitude-spectrum nuclear norm for low-rank amplitude recovery. We examine the power of these regularizers through experiments, which illustrates the utility of ERx.