# Mind the Gap: Norm-Aware Adaptive Robust Loss for Multivariate Least-Squares Problems

arXiv: https://arxiv.org/abs/2206.09215 · HTML: https://mlsanity.com/paper/2206.09215/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jun 2022.

## Built on by (most cited first, 5 of 5)

- [Estimation Contracts for Outlier-Robust Geometric Perception](https://mlsanity.com/paper/2208.10521/) (2022)
- [Bayesian Fields: Task-driven Open-Set Semantic Gaussian Splatting](https://mlsanity.com/paper/2503.05949/) (2025)
- [Adaptive Robust Estimator for Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2603.21574/) (2026)
- [SANDRO: a Robust Solver with a Splitting Strategy for Point Cloud Registration](https://mlsanity.com/paper/2503.07743/) (2025)
- [Dual-Channel Robust Group-Relative Policy Optimization via Advantage and Sequence-Weight Estimation](https://mlsanity.com/paper/2609.36944/) (2026)

## Abstract (arXiv)

Measurement outliers are unavoidable when solving real-world robot state estimation problems. A large family of robust loss functions (RLFs) exists to mitigate the effects of outliers, including newly developed adaptive methods that do not require parameter tuning. All of these methods assume that residuals follow a zero-mean Gaussian-like distribution. However, in multivariate problems the residual is often defined as a norm, and norms follow a Chi-like distribution with a non-zero mode value. This produces a "mode gap" that impacts the convergence rate and accuracy of existing RLFs. The proposed approach, "Adaptive MB," accounts for this gap by first estimating the mode of the residuals using an adaptive Chi-like distribution. Applying an existing adaptive weighting scheme only to residuals greater than the mode leads to more robust performance and faster convergence times in two fundamental state estimation problems, point cloud alignment and pose averaging.