# Deep Single-Index Fr'echet Regression

arXiv: https://arxiv.org/abs/2606.06957 · HTML: https://mlsanity.com/paper/2606.06957/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 49 references that are in the map.
- First posted on arXiv 5 Jun 2026.

## Builds on (most cited first, 20 of 20)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Fr'echet Regression for Random Objects with Euclidean Predictors](https://mlsanity.com/paper/1608.03012/) (2016)
- [Wasserstein Regression](https://mlsanity.com/paper/2006.09660/) (2020)
- [Network Regression with Graph Laplacians](https://mlsanity.com/paper/2109.02981/) (2021)
- [SPD-DDPM: Denoising Diffusion Probabilistic Models in the Symmetric Positive Definite Space](https://mlsanity.com/paper/2312.08200/) (2023)
- [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)
- [Wasserstein F-tests and Confidence Bands for the Fr`echet Regression of Density Response Curves](https://mlsanity.com/paper/1910.13418/) (2019)
- [Wasserstein Regression with Empirical Measures and Density Estimation for Sparse Data](https://mlsanity.com/paper/2308.12540/) (2023)
- [Regression with Distance Matrices](https://mlsanity.com/paper/1303.3750/) (2013)
- [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)
- [Sliced Wasserstein Regression](https://mlsanity.com/paper/2306.10601/) (2023)
- [Fr'echet single index models for object response regression](https://mlsanity.com/paper/2108.06058/) (2021)
- [Errors-in-variables Fr'echet Regression with Low-rank Covariate Approximation](https://mlsanity.com/paper/2305.09282/) (2023)
- [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)

## Abstract (arXiv)

Predicting outputs that are located in non-Euclidean spaces, such as probability distributions, networks, and symmetric positive-definite matrices, is becoming increasingly important in modern data analysis, particularly when inputs are high-dimensional. We propose DeSI (Deep Single-Index Fr'echet Regression), a semiparametric framework for regression with metric space-valued outputs and multivariate inputs that assumes a single-index structure for the conditional Fr'echet mean. DeSI estimates an interpretable index direction, which quantifies the relative importance of inputs, using a deep neural network, and performs Fr'echet regression along the resulting one-dimensional index in the target metric space. This structure mitigates the curse of dimensionality while retaining interpretability, which stands in contrast to standard deep neural networks. We establish theoretical guarantees for DeSI, including uniform approximation and convergence rates, and demonstrate its strong predictive performance through simulations on distributions, networks, and symmetric positive-definite matrices, as well as an application to compositional mood data from New Jersey.