Yangdi Jiang

2022 Edmonton Marathon

My research lies at the intersection of geometry, statistics, and machine learning. My past and ongoing work has focused on geometric statistics under privacy constraints, including differentially private Fréchet mean estimation, a central limit theorem for the differentially private Fréchet mean estimator, and Bayesian deconvolution on manifolds.

Recently, I have developed a growing interest in leveraging geometry and topology to understand the inner workings of AI and machine learning models. For instance, the parameter spaces of various neural networks exhibit singular geometric structures, and generalization error has been shown to correlate with the persistent homology dimension of their training trajectories.

When I’m not working, you’ll probably find me out running, anywhere from quiet neighbourhood streets to marathon courses.

selected publications

Gaussian differential privacy on Riemannian manifolds

Yangdi Jiang, Xiaotian Chang, Yi Liu, and 3 more authors

In Proceedings of the 37th International Conference on Neural Information Processing Systems, Dec 2023

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We develop an advanced approach for extending Gaussian Differential Privacy (GDP) to general Riemannian manifolds. The concept of GDP stands out as a prominent privacy definition that strongly warrants extension to manifold settings, due to its central limit properties. By harnessing the power of the renowned Bishop-Gromov theorem in geometric analysis, we propose a Riemannian Gaussian distribution that integrates the Riemannian distance, allowing us to achieve GDP in Riemannian manifolds with bounded Ricci curvature. To the best of our knowledge, this work marks the first instance of extending the GDP framework to accommodate general Riemannian manifolds, encompassing curved spaces, and circumventing the reliance on tangent space summaries. We provide a simple algorithm to evaluate the privacy budget μ on any one-dimensional manifold and introduce a versatile Markov Chain Monte Carlo (MCMC)-based algorithm to calculate μ on any Riemannian manifold with constant curvature. Through simulations on one of the most prevalent manifolds in statistics, the unit sphere Sd, we demonstrate the superior utility of our Riemannian Gaussian mechanism in comparison to the previously proposed Riemannian Laplace mechanism for implementing GDP.