# A Hypergradient Approach to Robust Regression without Correspondence

arXiv: https://arxiv.org/abs/2012.00123 · HTML: https://mlsanity.com/paper/2012.00123/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Nov 2020.

## Built on by (most cited first, 7 of 7)

- [FlashSinkhorn: IO-Aware Entropic Optimal Transport on GPU](https://mlsanity.com/paper/2602.03067/) (2026)
- [Unlabeled Principal Component Analysis and Matrix Completion](https://mlsanity.com/paper/2101.09446/) (2021)
- [Learning Graphon Autoencoders for Generative Graph Modeling](https://mlsanity.com/paper/2105.14244/) (2021)
- [Low-rank Matrix Recovery With Unknown Correspondence](https://mlsanity.com/paper/2110.07959/) (2021)
- [Shuffled linear regression through graduated convex relaxation](https://mlsanity.com/paper/2209.15608/) (2022)
- [Regularized Optimal Transport Layers for Generalized Global Pooling Operations](https://mlsanity.com/paper/2212.06339/) (2022)
- [Robust Graph Matching Using An Unbalanced Hierarchical Optimal Transport Framework](https://mlsanity.com/paper/2310.12081/) (2023)

## Abstract (arXiv)

We consider a variant of regression problem, where the correspondence between input and output data is not available. Such shuffled data is commonly observed in many real world problems. Taking flow cytometry as an example, the measuring instruments may not be able to maintain the correspondence between the samples and the measurements. Due to the combinatorial nature of the problem, most existing methods are only applicable when the sample size is small, and limited to linear regression models. To overcome such bottlenecks, we propose a new computational framework -- ROBOT -- for the shuffled regression problem, which is applicable to large data and complex nonlinear models. Specifically, we reformulate the regression without correspondence as a continuous optimization problem. Then by exploiting the interaction between the regression model and the data correspondence, we develop a hypergradient approach based on differentiable programming techniques. Such a hypergradient approach essentially views the data correspondence as an operator of the regression, and therefore allows us to find a better descent direction for the model parameter by differentiating through the data correspondence. ROBOT can be further extended to the inexact correspondence setting, where there may not be an exact alignment between the input and output data. Thorough numerical experiments show that ROBOT achieves better performance than existing methods in both linear and nonlinear regression tasks, including real-world applications such as flow cytometry and multi-object tracking.