# From matrix inversion to constraints: provably tighter confidence regions for importance weights in label shift

arXiv: https://arxiv.org/abs/2609.14802 · HTML: https://mlsanity.com/paper/2609.14802/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 42 references that are in the map.
- First posted on arXiv 13 Sep 2026.

## Builds on (most cited first, 26 of 26)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [MobileNetV2: Inverted Residuals and Linear Bottlenecks](https://mlsanity.com/paper/1801.04381/) (2018)
- [nuScenes: A multimodal dataset for autonomous driving](https://mlsanity.com/paper/1903.11027/) (2019)
- [Character-level Convolutional Networks for Text Classification](https://mlsanity.com/paper/1509.01626/) (2015)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Designing Network Design Spaces](https://mlsanity.com/paper/2003.13678/) (2020)
- [Conformal Prediction Under Covariate Shift](https://mlsanity.com/paper/1904.06019/) (2019)
- [Adaptive Conformal Inference Under Distribution Shift](https://mlsanity.com/paper/2106.00170/) (2021)
- [Detecting and Correcting for Label Shift with Black Box Predictors](https://mlsanity.com/paper/1802.03916/) (2018)
- [Regularized Learning for Domain Adaptation under Label Shifts](https://mlsanity.com/paper/1903.09734/) (2019)
- [A Unified View of Label Shift Estimation](https://mlsanity.com/paper/2003.07554/) (2020)
- [Distribution-free uncertainty quantification for classification under label shift](https://mlsanity.com/paper/2103.03323/) (2021)
- [PAC Prediction Sets Under Covariate Shift](https://mlsanity.com/paper/2106.09848/) (2021)
- [Online Adaptation to Label Distribution Shift](https://mlsanity.com/paper/2107.04520/) (2021)
- [N24News: A New Dataset for Multimodal News Classification](https://mlsanity.com/paper/2108.13327/) (2021)
- [Conformal Prediction for Federated Uncertainty Quantification Under Label Shift](https://mlsanity.com/paper/2306.05131/) (2023)
- [PAC Prediction Sets Under Label Shift](https://mlsanity.com/paper/2310.12964/) (2023)
- [The cost of using exact confidence intervals for a binomial proportion](https://mlsanity.com/paper/1303.1288/) (2013)
- [ELSA: Efficient Label Shift Adaptation through the Lens of Semiparametric Models](https://mlsanity.com/paper/2305.19123/) (2023)
- [Online Feature Updates Improve Online (Generalized) Label Shift Adaptation](https://mlsanity.com/paper/2402.03545/) (2024)
- [Classification under Nuisance Parameters and Generalized Label Shift in Likelihood-Free Inference](https://mlsanity.com/paper/2402.05330/) (2024)
- [Addressing Label Shift in Distributed Learning via Entropy Regularization](https://mlsanity.com/paper/2502.02544/) (2025)

## Abstract (arXiv)

Importance weights are essential in domain adaptation under label shift, yet their utility is often undermined by the finite sample uncertainty associated with their estimation. Existing methods typically analyze this uncertainty through Gaussian elimination on interval-valued linear systems, which leads to overly conservative confidence regions and inefficient downstream applications. We propose a paradigm shift from inversion-based inference to a direct matrix constraint framework. We use this framework to define a joint confidence region and extract marginal intervals via linear programming, deriving provably tighter bounds for importance weights while maintaining exact finite-sample validity. Furthermore, we analyze the confidence region's geometry and provide the theoretical results for its diameter bounds. Evaluated across text, image, multimodal benchmarks, including AGNews, MNIST, CIFAR-10, N24News, and a real-world autonomous driving dataset, nuImages, our approach consistently yields shorter confidence intervals and smaller prediction sets than inversion-based methods.