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

arXiv: https://arxiv.org/abs/2108.07406 · HTML: https://mlsanity.com/paper/2108.07406/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 41 references that are in the map.
- First posted on arXiv 17 Aug 2021.

## Builds on (most cited first, 13 of 13)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Image reconstruction by domain transform manifold learning](https://mlsanity.com/paper/1704.08841/) (2017)
- [Optimal rates for zero-order convex optimization: the power of two function evaluations](https://mlsanity.com/paper/1312.2139/) (2013)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Riemannian SVRG: Fast Stochastic Optimization on Riemannian Manifolds](https://mlsanity.com/paper/1605.07147/) (2016)
- [Multi-scale exploration of convex functions and bandit convex optimization](https://mlsanity.com/paper/1507.06580/) (2015)
- [Projection-Free Bandit Convex Optimization](https://mlsanity.com/paper/1805.07474/) (2018)
- [Improved Regret Bounds for Projection-free Bandit Convex Optimization](https://mlsanity.com/paper/1910.03374/) (2019)
- [Efficient Manifold and Subspace Approximations with Spherelets](https://mlsanity.com/paper/1706.08263/) (2017)
- [Regret Bounds without Lipschitz Continuity: Online Learning with Relative-Lipschitz Losses](https://mlsanity.com/paper/2010.12033/) (2020)
- [Bandit Convex Optimization: sqrt{T} Regret in One Dimension](https://mlsanity.com/paper/1502.06398/) (2015)
- [Geodesic Distance Estimation with Spherelets](https://mlsanity.com/paper/1907.00296/) (2019)
- [Classification via local manifold approximation](https://mlsanity.com/paper/1903.00985/) (2019)

## Built on by (most cited first, 6 of 6)

- [Stochastic Zeroth Order Gradient and Hessian Estimators: Variance Reduction and Refined Bias Bounds](https://mlsanity.com/paper/2205.14737/) (2022)
- [Zeroth-order Riemannian Averaging Stochastic Approximation Algorithms](https://mlsanity.com/paper/2309.14506/) (2023)
- [On Sharp Stochastic Zeroth Order Hessian Estimators over Riemannian Manifolds](https://mlsanity.com/paper/2201.10780/) (2022)
- [Convergence Rates of Stochastic Zeroth-order Gradient Descent for L ojasiewicz Functions](https://mlsanity.com/paper/2210.16997/) (2022)
- [Federated Learning on Riemannian Manifolds: A Gradient-Free Projection-Based Approach](https://mlsanity.com/paper/2507.22855/) (2025)
- [Riemannian Zeroth-Order Gradient Estimation with Structure-Preserving Metrics for Geodesically Incomplete Manifolds](https://mlsanity.com/paper/2601.08039/) (2026)

## Abstract (arXiv)

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.