Deep Single-Index Fréchet Regression

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.

Understanding how ordinary numerical inputs are related to complex outputs is an important but challenging problem. In this work, we propose DeSI, a method for settings where the input variables are standard numerical measurements, but the outputs have more complicated structures, such as distributions, networks or covariance matrices. DeSI uses a deep neural network to combine many input variables into one meaningful summary score, called a single index. This single index is then used to predict the complex output with a regression model designed for non-standard data. DeSI is especially useful in situations where a single-index structure is a reasonable assumption. Our method is accurate, computationally efficient, robust, and easier to interpret, because it reduces many input variables to one summary direction and provides known weights that show how the input variables contribute.