Deep Single-Index Fréchet Regression

Muqing Cui, Yidong Zhou, Su I Iao, Hans-Georg Müller
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:22095-22114, 2026.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-cui26l, title = {Deep Single-Index Fréchet Regression}, author = {Cui, Muqing and Zhou, Yidong and Iao, Su I and M\"{u}ller, Hans-Georg}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {22095--22114}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/cui26l/cui26l.pdf}, url = {https://proceedings.mlr.press/v306/cui26l.html}, 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.} }
Endnote
%0 Conference Paper %T Deep Single-Index Fréchet Regression %A Muqing Cui %A Yidong Zhou %A Su I Iao %A Hans-Georg Müller %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-cui26l %I PMLR %P 22095--22114 %U https://proceedings.mlr.press/v306/cui26l.html %V 306 %X 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.
APA
Cui, M., Zhou, Y., Iao, S.I. & Müller, H.. (2026). Deep Single-Index Fréchet Regression. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:22095-22114 Available from https://proceedings.mlr.press/v306/cui26l.html.

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