# Denoising Linear Models with Permuted Data

## TL;DR

This paper characterizes the minimax error rate for denoising in permuted linear models with Gaussian noise, analyzes efficient estimators, and provides algorithms applicable to image matching and datasets with outliers.

## Contribution

It offers a sharp characterization of the minimax error rate and analyzes the performance of efficient estimators for denoising in permuted linear models.

## Key findings

- Minimax error rate characterized up to logarithmic factors.
- Efficient estimators shown to be consistent across various parameters.
- Exact algorithm demonstrated on image point-cloud matching.

## Abstract

The multivariate linear regression model with shuffled data and additive Gaussian noise arises in various correspondence estimation and matching problems. Focusing on the denoising aspect of this problem, we provide a characterization the minimax error rate that is sharp up to logarithmic factors. We also analyze the performance of two versions of a computationally efficient estimator, and establish their consistency for a large range of input parameters. Finally, we provide an exact algorithm for the noiseless problem and demonstrate its performance on an image point-cloud matching task. Our analysis also extends to datasets with outliers.