# A characterization of proximity operators

arXiv: https://arxiv.org/abs/1807.04014 · HTML: https://mlsanity.com/paper/1807.04014/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jul 2018.

## Built on by (most cited first, 30 of 32)

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- [You KAN Do It in a Single Shot: Plug-and-Play Methods with Single-Instance Priors](https://mlsanity.com/paper/2412.06204/) (2024)
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- [Fluctuation-based deconvolution in fluorescence microscopy using plug-and-play denoisers](https://mlsanity.com/paper/2303.11212/) (2023)
- [Compressive Recovery of Sparse Precision Matrices](https://mlsanity.com/paper/2311.04673/) (2023)
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- [Controlled Learning of Pointwise Nonlinearities in Neural-Network-Like Architectures](https://mlsanity.com/paper/2408.13114/) (2024)
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- [Solving Imaging Inverse Problems Using Plug-and-Play Denoisers: Regularization and Optimization Perspectives](https://mlsanity.com/paper/2509.03475/) (2025)
- [HPPP: Halpern-type Preconditioned Proximal Point Algorithms and Applications to Image Restoration](https://mlsanity.com/paper/2407.13120/) (2024)
- [Deep learning methods for inverse problems using connections between proximal operators and Hamilton-Jacobi equations](https://mlsanity.com/paper/2512.23829/) (2025)
- [A New Convergence Analysis of Plug-and-Play Proximal Gradient Descent Under Prior Mismatch](https://mlsanity.com/paper/2601.09831/) (2026)
- [Learning Affine-Equivariant Proximal Operators](https://mlsanity.com/paper/2604.15556/) (2026)

## Abstract (arXiv)

We characterize proximity operators, that is to say functions that map a vector to a solution of a penalized least squares optimization problem. Proximity operators of convex penalties have been widely studied and fully characterized by Moreau. They are also widely used in practice with nonconvex penalties such as the ℓ 0 pseudo-norm, yet the extension of Moreau's characterization to this setting seemed to be a missing element of the literature. We characterize proximity operators of (convex or nonconvex) penalties as functions that are the subdifferential of some convex potential. This is proved as a consequence of a more general characterization of so-called Bregman proximity operators of possibly nonconvex penalties in terms of certain convex potentials. As a side effect of our analysis, we obtain a test to verify whether a given function is the proximity operator of some penalty, or not. Many well-known shrinkage operators are indeed confirmed to be proximity operators. However, we prove that windowed Group-LASSO and persistent empirical Wiener shrinkage -- two forms of so-called social sparsity shrinkage-- are generally not the proximity operator of any penalty; the exception is when they are simply weighted versions of group-sparse shrinkage with non-overlapping groups.