# Metric space valued Fr'echet regression

arXiv: https://arxiv.org/abs/2602.05225 · HTML: https://mlsanity.com/paper/2602.05225/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 43 references that are in the map.
- First posted on arXiv 5 Feb 2026.

## Builds on (most cited first, 15 of 15)

- [Fr'echet Regression for Random Objects with Euclidean Predictors](https://mlsanity.com/paper/1608.03012/) (2016)
- [Loss minimization and parameter estimation with heavy tails](https://mlsanity.com/paper/1307.1827/) (2013)
- [Universal Bayes consistency in metric spaces](https://mlsanity.com/paper/1906.09855/) (2019)
- [Hypothesis Testing For Network Data in Functional Neuroimaging](https://mlsanity.com/paper/1407.5525/) (2014)
- [Universal consistency and rates of convergence of multiclass prototype algorithms in metric spaces](https://mlsanity.com/paper/2010.00636/) (2020)
- [Network Regression with Graph Laplacians](https://mlsanity.com/paper/2109.02981/) (2021)
- [Modeling Network Populations via Graph Distances](https://mlsanity.com/paper/1904.07367/) (2019)
- [Fr'echet random forests for metric space valued regression with non euclidean predictors](https://mlsanity.com/paper/1906.01741/) (2019)
- [Random Forest Weighted Local Fr'echet Regression with Random Objects](https://mlsanity.com/paper/2202.04912/) (2022)
- [Manifold valued data analysis of samples of networks, with applications in corpus linguistics](https://mlsanity.com/paper/1902.08290/) (2019)
- [Uniform convergence of local Fr'echet regression, with applications to locating extrema and time warping for metric space valued trajectories](https://mlsanity.com/paper/2006.13548/) (2020)
- [Universal consistency of the k-NN rule in metric spaces and Nagata dimension](https://mlsanity.com/paper/2003.00894/) (2020)
- [Universal consistency of the k-NN rule in metric spaces and Nagata dimension. II](https://mlsanity.com/paper/2305.17282/) (2023)
- [Robust Estimation in metric spaces: Achieving Exponential Concentration with a Fr'echet Median](https://mlsanity.com/paper/2504.14161/) (2025)
- [Bayesian Multiple Multivariate Density-Density Regression](https://mlsanity.com/paper/2601.02640/) (2026)

## Abstract (arXiv)

We consider the problem of estimating the Fr'echet and conditional Fr'echet mean from data taking values in separable metric spaces. Unlike Euclidean spaces, where well-established methods are available, there is no practical estimator that works universally for all metric spaces. Therefore, we introduce a computable estimator for the Fr'echet mean based on random quantization techniques and establish its universal consistency across any separable metric spaces. Additionally, we propose another estimator for the conditional Fr'echet mean, leveraging data-driven partitioning and quantization, and demonstrate its universal consistency when the output space is any Banach space.