New DeSI framework for non-Euclidean regression tackles curse of dimensionality

DeSI (Deep Single-Index Fréchet Regression) is a new semiparametric framework designed for regression tasks where outputs reside in metric spaces like probability distributions, networks, or symmetric positive-definite matrices—common in modern data analysis but challenging for standard deep learning. The model assumes a single-index structure for the conditional Fréchet mean, meaning it projects high-dimensional inputs onto a single interpretable direction before performing Fréchet regression in the target space. This approach mitigates the curse of dimensionality while offering clear interpretability: the learned index direction quantifies the relative importance of each input feature. Unlike black-box neural networks, DeSI provides both strong predictive power and actionable insights into which variables drive predictions.

The paper provides theoretical guarantees, including uniform approximation rates, and validates performance across diverse simulations (distributions, networks, SPD matrices) and a real-world application to compositional mood data from New Jersey. DeSI consistently outperforms standard deep neural networks, especially in high-dimensional settings. This work is particularly relevant for fields like neuroscience (brain networks), finance (covariance matrices), and geostatistics (distributional data), where non-Euclidean outputs are ubiquitous. The combination of deep learning flexibility with interpretable single-index modeling opens new avenues for trusted AI in complex data domains.