# GeoERM: Geometry-Aware Multi-Task Representation Learning on Riemannian Manifolds

arXiv: https://arxiv.org/abs/2505.02972 · HTML: https://mlsanity.com/paper/2505.02972/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 33 of its 57 references that are in the map.
- First posted on arXiv 5 May 2025.

## Builds on (most cited first, 30 of 33)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Gradient Surgery for Multi-Task Learning](https://mlsanity.com/paper/2001.06782/) (2020)
- [A Survey on Multi-Task Learning](https://mlsanity.com/paper/1707.08114/) (2017)
- [Exploiting Shared Representations for Personalized Federated Learning](https://mlsanity.com/paper/2102.07078/) (2021)
- [DeCAF: A Deep Convolutional Activation Feature for Generic Visual Recognition](https://mlsanity.com/paper/1310.1531/) (2013)
- [Transfusion: Understanding Transfer Learning for Medical Imaging](https://mlsanity.com/paper/1902.07208/) (2019)
- [Few-Shot Learning via Learning the Representation, Provably](https://mlsanity.com/paper/2002.09434/) (2020)
- [Scaling and Benchmarking Self-Supervised Visual Representation Learning](https://mlsanity.com/paper/1905.01235/) (2019)
- [The Benefit of Multitask Representation Learning](https://mlsanity.com/paper/1505.06279/) (2015)
- [On the Theory of Transfer Learning: The Importance of Task Diversity](https://mlsanity.com/paper/2006.11650/) (2020)
- [Provable Meta-Learning of Linear Representations](https://mlsanity.com/paper/2002.11684/) (2020)
- [Transfer Learning for High-dimensional Linear Regression: Prediction, Estimation, and Minimax Optimality](https://mlsanity.com/paper/2006.10593/) (2020)
- [Learning Invariant Feature Spaces to Transfer Skills with Reinforcement Learning](https://mlsanity.com/paper/1703.02949/) (2017)
- [Transfer Learning under High-dimensional Generalized Linear Models](https://mlsanity.com/paper/2105.14328/) (2021)
- [How Fine-Tuning Allows for Effective Meta-Learning](https://mlsanity.com/paper/2105.02221/) (2021)
- [The Advantage of Conditional Meta-Learning for Biased Regularization and Fine-Tuning](https://mlsanity.com/paper/2008.10857/) (2020)
- [Adaptive and Robust Multi-Task Learning](https://mlsanity.com/paper/2202.05250/) (2022)
- [Learning from Similar Linear Representations: Adaptivity, Minimaxity, and Robustness](https://mlsanity.com/paper/2303.17765/) (2023)
- [Multilingual Knowledge Graph Completion via Ensemble Knowledge Transfer](https://mlsanity.com/paper/2010.03158/) (2020)
- [Predicting with Proxies: Transfer Learning in High Dimension](https://mlsanity.com/paper/1812.11097/) (2018)
- [Targeting Underrepresented Populations in Precision Medicine: A Federated Transfer Learning Approach](https://mlsanity.com/paper/2108.12112/) (2021)
- [Fast Rates by Transferring from Auxiliary Hypotheses](https://mlsanity.com/paper/1412.1619/) (2014)
- [Robust angle-based transfer learning in high dimensions](https://mlsanity.com/paper/2210.12759/) (2022)
- [Learning Discrete Distributions from Untrusted Batches](https://mlsanity.com/paper/1711.08113/) (2017)
- [Subspace Recovery from Heterogeneous Data with Non-isotropic Noise](https://mlsanity.com/paper/2210.13497/) (2022)
- [Doubly Robust Augmented Model Accuracy Transfer Inference with High Dimensional Features](https://mlsanity.com/paper/2208.05134/) (2022)
- [On the Sample Complexity of Adversarial Multi-Source PAC Learning](https://mlsanity.com/paper/2002.10384/) (2020)
- [Multitask Learning and Bandits via Robust Statistics](https://mlsanity.com/paper/2112.14233/) (2021)
- [Multi-source Learning via Completion of Block-wise Overlapping Noisy Matrices](https://mlsanity.com/paper/2105.10360/) (2021)
- [Do Outliers Ruin Collaboration?](https://mlsanity.com/paper/1805.04720/) (2018)

## Abstract (arXiv)

Multi-Task Learning (MTL) seeks to boost statistical power and learning efficiency by discovering structure shared across related tasks. State-of-the-art MTL representation methods, however, usually treat the latent representation matrix as a point in ordinary Euclidean space, ignoring its often non-Euclidean geometry, thus sacrificing robustness when tasks are heterogeneous or even adversarial. We propose GeoERM, a geometry-aware MTL framework that embeds the shared representation on its natural Riemannian manifold and optimizes it via explicit manifold operations. Each training cycle performs (i) a Riemannian gradient step that respects the intrinsic curvature of the search space, followed by (ii) an efficient polar retraction to remain on the manifold, guaranteeing geometric fidelity at every iteration. The procedure applies to a broad class of matrix-factorized MTL models and retains the same per-iteration cost as Euclidean baselines. Across a set of synthetic experiments with task heterogeneity and on a wearable-sensor activity-recognition benchmark, GeoERM consistently improves estimation accuracy, reduces negative transfer, and remains stable under adversarial label noise, outperforming leading MTL and single-task alternatives.