Non-Euclidean Data Analysis

From Novice to Expert — a course on Non-Euclidean Data Analysis.

This course is a self-contained introduction to the statistical analysis of data that live in non-Euclidean spaces — such as metric spaces, Riemannian manifolds, and Wasserstein spaces of probability distributions. Starting from first principles, it presents the theory of Fréchet means (existence, uniqueness, consistency, inference, and computation), builds up globally and locally weighted Fréchet regression, covers logistic regression with metric-space covariates, explores Wasserstein geometry and distribution-on-distribution regression, and culminates in a deep treatment of Riemannian manifolds and geodesic regression. Each lecture combines mathematical exposition with interactive visualizations and exercises, making the material accessible to advanced undergraduates, graduate students, and researchers alike.

1 Part I: Foundations

• Lecture 1: Introduction to Non-Euclidean Data Analysis

2 Part II: Fréchet Means in Metric Spaces

• Lecture 2: Fréchet Means — Existence and Uniqueness
• Lecture 3: Fréchet Means — Consistency and Convergence Rate
• Lecture 4: Fréchet Means — Inference in General Metric Spaces
• Lecture 5: Fréchet Means — Computation

3 Part III: Fréchet Regression

• Lecture 6: Fréchet Regression — Global Fréchet Regression
• Lecture 7: Fréchet Regression — Kernel Regression
• Lecture 8: Fréchet Regression — Local Fréchet Regression
• Lecture 9: Fréchet Regression — Total-Variation Regularized Fréchet Regression

4 Part IV: Logistic Regression with Metric-Space Covariates

• Lecture 10: Metric-Space Logistic Regression

5 Part V: Wasserstein Geometry and Distributional Data

• Lecture 11: The Wasserstein Distance and Its Geometry
• Lecture 12: Wasserstein Barycenters
• Lecture 13: Probability Distributions from Samples
• Lecture 14: Distribution-on-Distribution Regression

6 Part VI: Riemannian Manifolds

• Lecture 15: Riemannian Manifolds — Foundations
• Lecture 16: Fréchet Means on Riemannian Manifolds
• Lecture 17: Central Limit Theorems for Intrinsic Fréchet Means
• Lecture 18: Geodesic Regression on Riemannian Manifolds
• Lecture 19: Local Polynomial Regression on Manifolds

Desktop App

Prefer a standalone app? Download NEDA for your platform:

macOS .dmg installer Windows .zip archive Android .apk installer