# Elliptic Loss Regularization

arXiv: https://arxiv.org/abs/2503.02138 · HTML: https://mlsanity.com/paper/2503.02138/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 37 references that are in the map.
- First posted on arXiv 4 Mar 2025.

## Builds on (most cited first, 27 of 27)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [mixup: Beyond Empirical Risk Minimization](https://mlsanity.com/paper/1710.09412/) (2017)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [WILDS: A Benchmark of in-the-Wild Distribution Shifts](https://mlsanity.com/paper/2012.07421/) (2020)
- [Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization](https://mlsanity.com/paper/1911.08731/) (2019)
- [Deep CORAL: Correlation Alignment for Deep Domain Adaptation](https://mlsanity.com/paper/1607.01719/) (2016)
- [Out-of-Distribution Generalization via Risk Extrapolation (REx)](https://mlsanity.com/paper/2003.00688/) (2020)
- [MedMNIST v2 -- A large-scale lightweight benchmark for 2D and 3D biomedical image classification](https://mlsanity.com/paper/2110.14795/) (2021)
- [Just Train Twice: Improving Group Robustness without Training Group Information](https://mlsanity.com/paper/2107.09044/) (2021)
- [Gradient Matching for Domain Generalization](https://mlsanity.com/paper/2104.09937/) (2021)
- [Adversarial Domain Adaptation with Domain Mixup](https://mlsanity.com/paper/1912.01805/) (2019)
- [Invariance Principle Meets Information Bottleneck for Out-of-Distribution Generalization](https://mlsanity.com/paper/2106.06607/) (2021)
- [Improving Out-of-Distribution Robustness via Selective Augmentation](https://mlsanity.com/paper/2201.00299/) (2022)
- [How Does Mixup Help With Robustness and Generalization?](https://mlsanity.com/paper/2010.04819/) (2020)
- [Large-Scale Methods for Distributionally Robust Optimization](https://mlsanity.com/paper/2010.05893/) (2020)
- [On Mixup Regularization](https://mlsanity.com/paper/2006.06049/) (2020)
- [Regularizing Deep Networks with Semantic Data Augmentation](https://mlsanity.com/paper/2007.10538/) (2020)
- [The Effects of Regularization and Data Augmentation are Class Dependent](https://mlsanity.com/paper/2204.03632/) (2022)
- [Data augmentation instead of explicit regularization](https://mlsanity.com/paper/1806.03852/) (2018)
- [C-Mixup: Improving Generalization in Regression](https://mlsanity.com/paper/2210.05775/) (2022)
- [DORO: Distributional and Outlier Robust Optimization](https://mlsanity.com/paper/2106.06142/) (2021)
- [How Much Data Are Augmentations Worth? An Investigation into Scaling Laws, Invariance, and Implicit Regularization](https://mlsanity.com/paper/2210.06441/) (2022)
- [UMIX: Improving Importance Weighting for Subpopulation Shift via Uncertainty-Aware Mixup](https://mlsanity.com/paper/2209.08928/) (2022)
- [Sample Efficiency of Data Augmentation Consistency Regularization](https://mlsanity.com/paper/2202.12230/) (2022)
- [MixupE: Understanding and Improving Mixup from Directional Derivative Perspective](https://mlsanity.com/paper/2212.13381/) (2022)
- [Parabolic Continual Learning](https://mlsanity.com/paper/2503.02117/) (2025)

## Abstract (arXiv)

Regularizing neural networks is important for anticipating model behavior in regions of the data space that are not well represented. In this work, we propose a regularization technique for enforcing a level of smoothness in the mapping between the data input space and the loss value. We specify the level of regularity by requiring that the loss of the network satisfies an elliptic operator over the data domain. To do this, we modify the usual empirical risk minimization objective such that we instead minimize a new objective that satisfies an elliptic operator over points within the domain. This allows us to use existing theory on elliptic operators to anticipate the behavior of the error for points outside the training set. We propose a tractable computational method that approximates the behavior of the elliptic operator while being computationally efficient. Finally, we analyze the properties of the proposed regularization to understand the performance on common problems of distribution shift and group imbalance. Numerical experiments confirm the utility of the proposed regularization technique.