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

TL;DR

GeoERM introduces a geometry-aware multi-task learning framework that embeds shared representations on Riemannian manifolds, improving robustness and accuracy especially in heterogeneous and adversarial settings.

Contribution

It is the first to incorporate Riemannian geometry into multi-task learning for matrix-factorized models, ensuring geometric fidelity during optimization.

Findings

Consistently improves estimation accuracy over Euclidean methods.

Reduces negative transfer in multi-task learning.

Remains stable under adversarial label noise.

Abstract

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.