# Dynamic Frechet Regression with Feature Selection for Distributional Data

arXiv: https://arxiv.org/abs/2607.09613 · HTML: https://mlsanity.com/paper/2607.09613/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 27 references that are in the map.
- First posted on arXiv 10 Jul 2026.

## Builds on (most cited first, 9 of 9)

- [Fr'echet Regression for Random Objects with Euclidean Predictors](https://mlsanity.com/paper/1608.03012/) (2016)
- [Functional Regression](https://mlsanity.com/paper/1406.4068/) (2014)
- [Wasserstein Autoregressive Models for Density Time Series](https://mlsanity.com/paper/2006.12640/) (2020)
- [Regression with Distance Matrices](https://mlsanity.com/paper/1303.3750/) (2013)
- [Dimension Reduction for Fr'echet Regression](https://mlsanity.com/paper/2110.00467/) (2021)
- [Nonparametric and Regularized Dynamical Wasserstein Barycenters for Sequential Observations](https://mlsanity.com/paper/2210.01918/) (2022)
- [Deep Fr'echet Regression](https://mlsanity.com/paper/2407.21407/) (2024)
- [Frequentist Model Averaging for Global Fr'echet Regression](https://mlsanity.com/paper/2309.01691/) (2023)
- [Kernel Regression with Sparse Metric Learning](https://mlsanity.com/paper/1712.09001/) (2017)

## Abstract (arXiv)

Many scientific and engineering applications generate responses that are not scalars or vectors, but statistical objects whose form evolves over an ordered index such as time, depth. Probability distributions are a prominent example, capturing variability and uncertainty that cannot be summarized by low-dimensional statistics. When such responses are observed sequentially, the resulting dynamic distributional trajectories pose significant challenges for regression, particularly in relating scalar predictors to both within-index variability and cross-index evolution. We propose Dynamic Fr'echet Regression (DFR), a framework for modeling index-dependent trajectories of distribution-valued responses. DFR extends Global Fr'echet Regression by introducing an index-aware weighting mechanism. At each index, predictions are defined as weighted Fr'echet means in a metric space of distributions (e.g., Wasserstein space), preserving the intrinsic geometry of the response. The weights depend jointly on predictor similarity and index proximity, enabling index-specific prediction while borrowing strength across neighboring indices. To improve interpretability in high-dimensional settings, DFR incorporates a geometry-aware feature selection approach based on sparse metric learning, which identifies predictors driving distributional dynamics without relying on Euclidean coefficients. Simulation studies show improved predictive accuracy and feature recovery over existing methods. An application to additive manufacturing data demonstrates its ability to produce interpretable, index-specific distributional predictions.