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Ridge Regression on Riemannian Manifolds for Time-Series Prediction

Esfandiar Nava-Yazdani

TL;DR

This work addresses predicting manifold-valued time series by extending ridge regression to arbitrary Riemannian manifolds through best-fitting Bézier curves and Mahalanobis regularization. It derives an explicit gradient for the penalized objective using adjoint differentials, enabling efficient Riemannian gradient descent, and leverages the de Casteljau-based evaluation for fast computation. The approach is validated on synthetic spherical trajectories, showing substantial error reductions over unregularized regression, and applied to hurricane forecasting with competitive 12-hour forecasts using only historical data, highlighting sample efficiency. The contributions include (i) a natural intrinsic extension of ridge regression to manifolds, (ii) a closed-form gradient for the regularized objective, (iii) a scalable Bézier-polynomial regression framework, and (iv) broad applicability to trajectory modeling in domains such as meteorology, robotics, and medical imaging.

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 Bézier 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.

Ridge Regression on Riemannian Manifolds for Time-Series Prediction

TL;DR

This work addresses predicting manifold-valued time series by extending ridge regression to arbitrary Riemannian manifolds through best-fitting Bézier curves and Mahalanobis regularization. It derives an explicit gradient for the penalized objective using adjoint differentials, enabling efficient Riemannian gradient descent, and leverages the de Casteljau-based evaluation for fast computation. The approach is validated on synthetic spherical trajectories, showing substantial error reductions over unregularized regression, and applied to hurricane forecasting with competitive 12-hour forecasts using only historical data, highlighting sample efficiency. The contributions include (i) a natural intrinsic extension of ridge regression to manifolds, (ii) a closed-form gradient for the regularized objective, (iii) a scalable Bézier-polynomial regression framework, and (iv) broad applicability to trajectory modeling in domains such as meteorology, robotics, and medical imaging.

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 Bézier 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.

Paper Structure

This paper contains 19 sections, 1 theorem, 23 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Prediction for Trajectories in Riemannian Manifolds
  3. Bézier Polynomials and Regression
  4. De Casteljau Algorithm
  5. Regression Model
  6. Ridge Regression
  7. Linearization and Gradient
  8. Mahalanobis Distance and Fitting Data
  9. Parameter Optimization and Forecasting
  10. Numerical Experiments
  11. Synthetic Validation
  12. Application: Hurricane Forecasting
  13. Dataset
  14. Experiments and Discussion
  15. Conclusions
  16. ...and 4 more sections

Key Result

proposition thmcounterproposition

Consider the function $f$ defined on $\mathcal{U}$ by Fix $x\in \mathcal{U}$. Let $\gamma$ be the geodesic given by $\gamma (t)=\mathrm{Exp}_\mu (t\mathrm{Log}_\mu x)$ with $t\in I$. Then where the superscript $^\dagger$ represents the adjoint. Moreover, let $k$ be the function defined by $k(x)=\mathrm{Exp}_\mu (W\mathrm{Log}_\mu x)$ on $\mathcal{U}$, and denote the tangent map of $k$ evaluated

Figures (5)

  • Figure 1: Construction of a quadratic Bézier curve on sphere via the de Casteljau algorithm. Control points $b_1, b_2, b_3$ (red) determine the curve (black). For parameter $t\in I$, we recursively interpolate: connect $b_1,b_2$ and $b_2,b_3$ with geodesics (blue), evaluate at $t$ to get intermediate points (green), then connect these with another geodesic and evaluate at t to obtain the curve point $p(t;b)$. This geometric construction generalizes naturally to higher degrees and arbitrary manifolds.
  • Figure 2: Two Representative forecasts from synthetic experiments, illustrating the prediction of the ridge regression (red) compared to the ground truth (blue). The close agreement demonstrates effective regularization.
  • Figure 3: 2021 Atlantic tracks and their intensities (maximum wind speeds).
  • Figure 4: Two hurricane tracks (blue) with their Bézier polynomial representations (red) using $n=6$ control points. High $R^2$ values indicate the polynomial model captures trajectory geometry effectively. These successful fits represent typical cases where our method performs well.
  • Figure 5: Challenging forecast case: 12 h forecasts (red) via ridge regression for the category 4 and long-lasting 2021 anomalous hurricane Sam (blue). This hurricane has an anomalous highly complex dynamics, and very often and rapidly changes its intensity.

Theorems & Definitions (2)

  • proposition thmcounterproposition: Gradient of squared Mahalanobis distance
  • proof