On Measures of Robustness for Statistical Procedures on Non-Euclidean Spaces

• Department of Statistics & Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada [schenouri@uwaterloo.ca]

Keywords: Manifolds – Influence function – Breakdown value – dimensionality reduction – categorical data analysis

Abstract

Statistical robustness refers to the resilience of a statistical procedure to deviations from underlying assumptions, such as outliers in the data, model misspecifications, and distributional contaminations. Classical robustness measures—such as qualitative robustness, influence functions, and breakdown values—were primarily developed for data and parameters in standard Euclidean spaces. However, data analyses increasingly involve complex structures with non-Euclidean geometry and manifolds, such as shapes, networks, trees, matrices, and function spaces. Practical examples include MRI scans, geometric measurements of additive manufacturing products, robotics, diffusion tensor imaging, geochemical compositions, face recognition, and various functional data.

Analyzing the aforementioned non-Euclidean data presents unique challenges, particularly regarding statistical robustness. While classical robustness measures (e.g., influence functions and breakdown values) offer guiding principles, applying them to non-Euclidean spaces requires reformulations that respect local geometry, geodesic distances, and topological nuances. This issue is even more pronounced when dealing with compact manifolds, where there is no natural notion of infinitesimal or extreme contaminations due to their boundedness. So, claiming robustness without a clear understanding of how to measure it in non-Euclidean spaces is problematic. Without a theoretical foundation, it is unclear whether the observed robustness based on a simulation study or intuition is intrinsic to the procedure or merely an artifact of the test cases. To rigorously assert that a procedure is robust in non-Euclidean spaces, one needs appropriate measures of robustness tailored to these spaces.

In this talk, we discuss the key hurdles in extending robust classical measures to non-Euclidean spaces, where standard linear notions of distance and outlyingness are not readily available. We provide some solutions and discuss examples, such as dimensionality reduction and categorical data analysis as special cases on compact manifolds.