# The Star Geometry of Critic-Based Regularizer Learning

arXiv: https://arxiv.org/abs/2408.16852 · HTML: https://mlsanity.com/paper/2408.16852/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 93 references that are in the map.
- First posted on arXiv 29 Aug 2024.

## Builds on (most cited first, 30 of 46)

- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Robust Compressed Sensing MRI with Deep Generative Priors](https://mlsanity.com/paper/2108.01368/) (2021)
- [Stochastic model-based minimization of weakly convex functions](https://mlsanity.com/paper/1803.06523/) (2018)
- [Solving Linear Inverse Problems Provably via Posterior Sampling with Latent Diffusion Models](https://mlsanity.com/paper/2307.00619/) (2023)
- [Score-Based Diffusion Models as Principled Priors for Inverse Imaging](https://mlsanity.com/paper/2304.11751/) (2023)
- [Ambient Diffusion: Learning Clean Distributions from Corrupted Data](https://mlsanity.com/paper/2305.19256/) (2023)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [Invertible generative models for inverse problems: mitigating representation error and dataset bias](https://mlsanity.com/paper/1905.11672/) (2019)
- [Phase Retrieval Under a Generative Prior](https://mlsanity.com/paper/1807.04261/) (2018)
- [Learned convex regularizers for inverse problems](https://mlsanity.com/paper/2008.02839/) (2020)
- [Uniform Convergence of Interpolators: Gaussian Width, Norm Bounds, and Benign Overfitting](https://mlsanity.com/paper/2106.09276/) (2021)
- [Total Deep Variation for Linear Inverse Problems](https://mlsanity.com/paper/2001.05005/) (2020)
- [Measuring Robustness in Deep Learning Based Compressive Sensing](https://mlsanity.com/paper/2102.06103/) (2021)
- [Sparse Modeling for Image and Vision Processing](https://mlsanity.com/paper/1411.3230/) (2014)
- [A Neural-Network-Based Convex Regularizer for Inverse Problems](https://mlsanity.com/paper/2211.12461/) (2022)
- [Blind Image Deconvolution using Deep Generative Priors](https://mlsanity.com/paper/1802.04073/) (2018)
- [Simple, Efficient, and Neural Algorithms for Sparse Coding](https://mlsanity.com/paper/1503.00778/) (2015)
- [Low-rank matrix recovery with composite optimization: good conditioning and rapid convergence](https://mlsanity.com/paper/1904.10020/) (2019)
- [Learning Weakly Convex Regularizers for Convergent Image-Reconstruction Algorithms](https://mlsanity.com/paper/2308.10542/) (2023)
- [Sensing Theorems for Unsupervised Learning in Linear Inverse Problems](https://mlsanity.com/paper/2203.12513/) (2022)
- [A note on the sample complexity of the Er-SpUD algorithm by Spielman, Wang and Wright for exact recovery of sparsely used dictionaries](https://mlsanity.com/paper/1601.02049/) (2016)
- [Learning the optimal Tikhonov regularizer for inverse problems](https://mlsanity.com/paper/2106.06513/) (2021)
- [What's in a Prior? Learned Proximal Networks for Inverse Problems](https://mlsanity.com/paper/2310.14344/) (2023)
- [Learning Sparsely Used Overcomplete Dictionaries via Alternating Minimization](https://mlsanity.com/paper/1310.7991/) (2013)
- [Dictionary Learning and Tensor Decomposition via the Sum-of-Squares Method](https://mlsanity.com/paper/1407.1543/) (2014)
- [Unsupervised Learning From Incomplete Measurements for Inverse Problems](https://mlsanity.com/paper/2201.12151/) (2022)

## Built on by (most cited first, 3 of 3)

- [Finite-Time Bounds for Average-Reward Fitted Q-Iteration](https://mlsanity.com/paper/2510.17391/) (2025)
- [Optimal Regularization Under Uncertainty: Distributional Robustness and Convexity Constraints](https://mlsanity.com/paper/2510.03464/) (2025)
- [Riemannian Archetypal Analysis: Interpretable non-linear data analysis on deformed star distributions](https://mlsanity.com/paper/2605.24113/) (2026)

## Abstract (arXiv)

Variational regularization is a classical technique to solve statistical inference tasks and inverse problems, with modern data-driven approaches parameterizing regularizers via deep neural networks showcasing impressive empirical performance. Recent works along these lines learn task-dependent regularizers. This is done by integrating information about the measurements and ground-truth data in an unsupervised, critic-based loss function, where the regularizer attributes low values to likely data and high values to unlikely data. However, there is little theory about the structure of regularizers learned via this process and how it relates to the two data distributions. To make progress on this challenge, we initiate a study of optimizing critic-based loss functions to learn regularizers over a particular family of regularizers: gauges (or Minkowski functionals) of star-shaped bodies. This family contains regularizers that are commonly employed in practice and shares properties with regularizers parameterized by deep neural networks. We specifically investigate critic-based losses derived from variational representations of statistical distances between probability measures. By leveraging tools from star geometry and dual Brunn-Minkowski theory, we illustrate how these losses can be interpreted as dual mixed volumes that depend on the data distribution. This allows us to derive exact expressions for the optimal regularizer in certain cases. Finally, we identify which neural network architectures give rise to such star body gauges and when do such regularizers have favorable properties for optimization. More broadly, this work highlights how the tools of star geometry can aid in understanding the geometry of unsupervised regularizer learning.