Deep Single-Index Fréchet Regression

2026-06-05

AI Summary

This work addresses the challenge of regression prediction in non-Euclidean spaces—such as probability distributions, networks, and symmetric positive-definite matrices—under high-dimensional inputs. The authors propose the DeSI framework, which integrates deep neural networks with single-index modeling to perform Fréchet regression along an interpretable one-dimensional direction within the target metric space. By doing so, DeSI mitigates the curse of dimensionality while preserving model interpretability, thereby overcoming the black-box nature of conventional deep networks and offering theoretical guarantees. Empirical evaluations demonstrate that DeSI achieves superior performance across diverse non-Euclidean data types and has been successfully applied to analyzing the emotional composition of New Jersey residents.

Abstract

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 Fréchet Regression), a semiparametric framework for regression with metric space-valued outputs and multivariate inputs that assumes a single-index structure for the conditional Fréchet mean. DeSI estimates an interpretable index direction, which quantifies the relative importance of inputs, using a deep neural network, and performs Fréchet 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.

Problem

• non-Euclidean regression
• high-dimensional inputs
• Fréchet mean
• metric space-valued outputs
• single-index model

Innovation

• Deep Single-Index
• Fréchet Regression
• Metric Space-valued Output
• Interpretable Direction
• Curse of Dimensionality