Publication

Data-Driven Regularization with Weak Convexity for Robust Image Reconstruction

2026

Abstract

We describe a practical framework for data-driven regularization in image reconstruction. It combines model expressivity with the guarantees of variational methods. The approach is based on a weakly convex ridge regularizer, defined as the composition of a convolutional filter bank and pointwise potentials constrained to be weakly convex. These potentials are implemented as learnable splines with a strict control of their weak-convexity modulus. The resulting denoisers outperform classic convex regularization techniques as well as competitive benchmarks such as BM3D, while they still correspond to the minimization of a convex energy. The learned regularizers further extend to general inverse problems with provable convergence to critical points. Overall, this framework shows that a controlled relaxation of convexity enables the design of learnable priors that achieve strong empirical performance while preserving mathematical guarantees.

Official source

https://infoscience.epfl.ch/entities/publication/7978c7b4-354d-4006-bb28-5e1a6885cb1e

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2026

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/7978c7b4-354d-4006-bb28-5e1a6885cb1e

About this result

Related concepts (32)

Convex function

In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain.

Uniformly convex space

In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. The concept of uniform convexity was first introduced by James A. Clarkson in 1936. A uniformly convex space is a normed vector space such that, for every there is some such that for any two vectors with and the condition implies that: Intuitively, the center of a line segment inside the unit ball must lie deep inside the unit ball unless the segment is short.

Convex hull

In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact.

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Data-Driven Regularization with Weak Convexity for Robust Image Reconstruction

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