On Implicit Concave Structures in Half-Quadratic Methods for Signal Reconstruction
Vittorio Latorre
TL;DR
This work defines implicit concave functions f(x)=V(Φ(x)) and develops a Fenchel-augmented problem L(x,σ)=⟨Φ(x),σ⟩−V^*(σ) that establishes a one-to-one correspondence between stationary points of f and L. The augmented problem is convex in the new variables σ and, in some cases, biconvex in (x,σ), enabling efficient block-coordinate descent methods. The authors prove that local optimality conditions (SONC/SOSC) carry over to the augmented formulation and show boundedness of L when f is bounded below. Applying the framework to edge-preserving regularizers via half-quadratic regularization, ψ(t)=V(t^2) yields a biconvex, bounded below augmented problem L_ep(x,σ) that is solvable by simple convex subproblems, offering a practical pathway for structured non-convex signal and image reconstruction problems.
Abstract
In this work, we introduce a new class of non-convex functions, called implicit concave functions, which are compositions of a concave function with a continuously differentiable mapping. We analyze the properties of their minimization by leveraging Fenchel conjugate theory to construct an augmented optimization problem. This reformulation yields a one-to-one correspondence between the stationary points and local minima of the original and augmented problems. Crucially, the augmented problem admits a natural variable splitting that reveals convexity with respect to at least one block, and, in some cases, leading to a biconvex structure that is more amenable to optimization. This enables the use of efficient block coordinate descent algorithms for solving otherwise non-convex problems. As a representative application, we show how this framework applies to half-quadratic regularization in signal reconstruction and image processing. We demonstrate that common edge-preserving regularizers fall within the proposed class, and that their corresponding augmented problems are biconvex and bounded from below. Our results offer both a theoretical foundation and a practical pathway for solving a broad class of structured non-convex problems.