# Metric Statistics: Exploration and Inference for Random Objects With Distance Profiles

arXiv: https://arxiv.org/abs/2202.06117 · HTML: https://mlsanity.com/paper/2202.06117/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Feb 2022.

## Built on by (most cited first, 4 of 4)

- [DFNN: A Deep Fr'echet Neural Network Framework for Learning Metric-Space-Valued Responses](https://mlsanity.com/paper/2510.17072/) (2025)
- [Geometric Renyi Differential Privacy: Ricci Curvature Characterized by Heat Diffusion Mechanisms](https://mlsanity.com/paper/2604.20761/) (2026)
- [Fr'echet Geodesic Boosting](https://mlsanity.com/paper/2509.18013/) (2025)
- [Distance Profile Embedding for Independence and Conditional Independence Testing of Random Objects](https://mlsanity.com/paper/2607.28981/) (2026)

## Abstract (arXiv)

This article provides an overview on the statistical modeling of complex data as increasingly encountered in modern data analysis. It is argued that such data can often be described as elements of a metric space that satisfies certain structural conditions and features a probability measure. We refer to the random elements of such spaces as random objects and to the emerging field that deals with their statistical analysis as metric statistics. Metric statistics provides methodology, theory and visualization tools for the statistical description, quantification of variation, centrality and quantiles, regression and inference for populations of random objects, inferring these quantities from available data and samples. In addition to a brief review of current concepts, we focus on distance profiles as a major tool for object data in conjunction with the pairwise Wasserstein transports of the underlying one-dimensional distance distributions. These pairwise transports lead to the definition of intuitive and interpretable notions of transport ranks and transport quantiles as well as two-sample inference. An associated profile metric complements the original metric of the object space and may reveal important features of the object data in data analysis. We demonstrate these tools for the analysis of complex data through various examples and visualizations.