# Fr'echet Regression for Random Objects with Euclidean Predictors

arXiv: https://arxiv.org/abs/1608.03012 · HTML: https://mlsanity.com/paper/1608.03012/

## Key facts

- Cited by 57 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Aug 2016.

## Built on by (most cited first, 30 of 57)

- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [SPD-DDPM: Denoising Diffusion Probabilistic Models in the Symmetric Positive Definite Space](https://mlsanity.com/paper/2312.08200/) (2023)
- [Random Forest Weighted Local Fr'echet Regression with Random Objects](https://mlsanity.com/paper/2202.04912/) (2022)
- [Lifting Weak Supervision To Structured Prediction](https://mlsanity.com/paper/2211.13375/) (2022)
- [Dimension Reduction for Fr'echet Regression](https://mlsanity.com/paper/2110.00467/) (2021)
- [Nonlinear Sufficient Dimension Reduction for Distribution-on-Distribution Regression](https://mlsanity.com/paper/2207.04613/) (2022)
- [Mitigating Discrimination in Insurance with Wasserstein Barycenters](https://mlsanity.com/paper/2306.12912/) (2023)
- [Medoid splits for efficient random forests in metric spaces](https://mlsanity.com/paper/2306.17031/) (2023)
- [Errors-in-variables Fr'echet Regression with Low-rank Covariate Approximation](https://mlsanity.com/paper/2305.09282/) (2023)
- [Geometry-Aware Adaptation for Pretrained Models](https://mlsanity.com/paper/2307.12226/) (2023)
- [Uncertainty quantification in metric spaces](https://mlsanity.com/paper/2405.05110/) (2024)
- [Deep Fr'echet Regression](https://mlsanity.com/paper/2407.21407/) (2024)
- [On Conditional Stochastic Interpolation for Generative Nonlinear Sufficient Dimension Reduction](https://mlsanity.com/paper/2512.18971/) (2025)
- [Functional additive models on manifolds of planar shapes and forms](https://mlsanity.com/paper/2109.02624/) (2021)
- [Theoretical analysis and computation of the sample Frechet mean for sets of large graphs based on spectral information](https://mlsanity.com/paper/2201.05923/) (2022)
- [Interpretable Causal Inference for Analyzing Wearable, Sensor, and Distributional Data](https://mlsanity.com/paper/2312.10569/) (2023)
- [Wasserstein Transfer Learning](https://mlsanity.com/paper/2505.17404/) (2025)
- [Conformal and kNN Predictive Uncertainty Quantification Algorithms in Metric Spaces](https://mlsanity.com/paper/2507.15741/) (2025)
- [End-to-End Deep Learning for Predicting Metric Space-Valued Outputs](https://mlsanity.com/paper/2509.23544/) (2025)
- [DFNN: A Deep Fr'echet Neural Network Framework for Learning Metric-Space-Valued Responses](https://mlsanity.com/paper/2510.17072/) (2025)
- [Geometry-Aware Uncertainty Quantification via Conformal Prediction on Manifolds](https://mlsanity.com/paper/2602.16015/) (2026)
- [Measure-to-measure Regression with Transformers](https://mlsanity.com/paper/2605.28075/) (2026)
- [Fisher-Pitman permutation tests based on nonparametric Poisson mixtures with application to single cell genomics](https://mlsanity.com/paper/2106.03022/) (2021)
- [Gaussian Process Subspace Regression for Model Reduction](https://mlsanity.com/paper/2107.04668/) (2021)
- [Conformal inference for regression on Riemannian Manifolds](https://mlsanity.com/paper/2310.08209/) (2023)
- [An Optimal Transport Approach for Network Regression](https://mlsanity.com/paper/2406.12204/) (2024)
- [When does the mean network capture the topology of a sample of networks?](https://mlsanity.com/paper/2408.03461/) (2024)
- [On metric choice in dimension reduction for Fr'echet regression](https://mlsanity.com/paper/2410.01783/) (2024)
- [Fr'echet Cumulative Covariance Net for Deep Nonlinear Sufficient Dimension Reduction with Random Objects](https://mlsanity.com/paper/2502.15374/) (2025)
- [Multivariate Uncertainty Quantification with Tomographic Quantile Forests](https://mlsanity.com/paper/2512.16383/) (2025)

## Abstract (arXiv)

Increasingly, statisticians are faced with the task of analyzing complex data that are non-Euclidean and specifically do not lie in a vector space. To address the need for statistical methods for such data, we introduce the concept of Fr'echet regression. This is a general approach to regression when responses are complex random objects in a metric space and predictors are in Rp, achieved by extending the classical concept of a Fr'echet mean to the notion of a conditional Fr'echet mean. We develop generalized versions of both global least squares regression and local weighted least squares smoothing. The target quantities are appropriately defined population versions of global and local regression for response objects in a metric space. We derive asymptotic rates of convergence for the corresponding fitted regressions using observed data to the population targets under suitable regularity conditions by applying empirical process methods. For the special case of random objects that reside in a Hilbert space, such as regression models with vector predictors and functional data as responses, we obtain a limit distribution. The proposed methods have broad applicability. Illustrative examples include responses that consist of probability distributions and correlation matrices, and we demonstrate both global and local Fr'echet regression for demographic and brain imaging data. Local Fr'echet regression is also illustrated via a simulation with response data which lie on the sphere.