From the Greene--Wu Convolution to Gradient Estimation over Riemannian Manifolds

Abstract

Over a complete Riemannian manifold of finite dimension, Greene and Wu introduced a convolution, known as Greene-Wu (GW) convolution. In this paper, we study properties of the GW convolution and apply it to non-Euclidean machine learning problems. In particular, we derive a new formula for how the curvature of the space would affect the curvature of the function through the GW convolution. Also, following the study of the GW convolution, a new method for gradient estimation over Riemannian manifolds is introduced.

References in corpus (9)

• Image reconstruction by domain transform manifold learning
• Neural Tangent Kernel: Convergence and Generalization in Neural Networks
• Riemannian SVRG: Fast Stochastic Optimization on Riemannian Manifolds
• Projection-free Online Learning
• Projection-Free Bandit Convex Optimization
• Geodesic Distance Estimation with Spherelets
• Efficient Manifold and Subspace Approximations with Spherelets
• Improved Regret Bounds for Projection-free Bandit Convex Optimization
• Classification via local manifold approximation

Cited by in corpus (1)

• Stochastic Zeroth Order Gradient and Hessian Estimators: Variance Reduction and Refined Bias Bounds