Ridge Regression on Riemannian Manifolds for Time-Series Prediction

TL;DR

This paper extends ridge regression to Riemannian manifolds for time-series prediction, utilizing geometric methods and efficient optimization, with successful validation on synthetic and real-world data.

Contribution

It introduces a novel intrinsic ridge regression framework on Riemannian manifolds, combining geometric fitting, covariance estimation, and Mahalanobis regularization with explicit gradient formulas.

Findings

Significant error reduction in synthetic spherical experiments

Effective hurricane forecasting results

Efficient numerical optimization via Riemannian gradient descent

Abstract

We propose a natural intrinsic extension of ridge regression from Euclidean spaces to general Riemannian manifolds for time-series prediction. Our approach combines Riemannian least-squares fitting via B\'ezier curves, empirical covariance on manifolds, and Mahalanobis distance regularization. A key technical contribution is an explicit formula for the gradient of the objective function using adjoint differentials, enabling efficient numerical optimization via Riemannian gradient descent. We validate our framework through synthetic spherical experiments (achieving significant error reduction over unregularized regression) and hurricane forecasting.