Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sufficient dimension reduction for regression with metric space-valued responses

Abdul-Nasah Soale, Yuexiao Dong

TL;DR

This work tackles regression with non-Euclidean, metric-space responses by replacing kernel-based Fréchet embedding with a fast, kernel-free Euclidean embedding that uses pairwise distances. It introduces ensemble-based surrogate responses and develops surrogate-assisted OLS and SIR estimators to recover the Fréchet central space ${\\mathcal S}_{Y|{\\bf X}}$ without requiring the response space to embed isometrically into a Hilbert space. The authors demonstrate through simulations and real-data analyses (COVID-19 transmission distributions and brain connectivity) that the surrogate-assisted methods often outperform kernel Fréchet counterparts, offering robustness to outliers and heteroscedasticity and enabling richer, distribution- and network-level insights. The approach provides a practical, scalable framework for dimension reduction with complex responses and suggests avenues for extending SDR to nonlinear settings and other SDR families.

Abstract

Data visualization and dimension reduction for regression between a general metric space-valued response and Euclidean predictors is proposed. Current Fréchét dimension reduction methods require that the response metric space be continuously embeddable into a Hilbert space, which imposes restriction on the type of metric and kernel choice. We relax this assumption by proposing a Euclidean embedding technique which avoids the use of kernels. Under this framework, classical dimension reduction methods such as ordinary least squares and sliced inverse regression are extended. An extensive simulation experiment demonstrates the superior performance of the proposed method on synthetic data compared to existing methods where applicable. The real data analysis of factors influencing the distribution of COVID-19 transmission in the U.S. and the association between BMI and structural brain connectivity of healthy individuals are also investigated.

Sufficient dimension reduction for regression with metric space-valued responses

TL;DR

This work tackles regression with non-Euclidean, metric-space responses by replacing kernel-based Fréchet embedding with a fast, kernel-free Euclidean embedding that uses pairwise distances. It introduces ensemble-based surrogate responses and develops surrogate-assisted OLS and SIR estimators to recover the Fréchet central space without requiring the response space to embed isometrically into a Hilbert space. The authors demonstrate through simulations and real-data analyses (COVID-19 transmission distributions and brain connectivity) that the surrogate-assisted methods often outperform kernel Fréchet counterparts, offering robustness to outliers and heteroscedasticity and enabling richer, distribution- and network-level insights. The approach provides a practical, scalable framework for dimension reduction with complex responses and suggests avenues for extending SDR to nonlinear settings and other SDR families.

Abstract

Data visualization and dimension reduction for regression between a general metric space-valued response and Euclidean predictors is proposed. Current Fréchét dimension reduction methods require that the response metric space be continuously embeddable into a Hilbert space, which imposes restriction on the type of metric and kernel choice. We relax this assumption by proposing a Euclidean embedding technique which avoids the use of kernels. Under this framework, classical dimension reduction methods such as ordinary least squares and sliced inverse regression are extended. An extensive simulation experiment demonstrates the superior performance of the proposed method on synthetic data compared to existing methods where applicable. The real data analysis of factors influencing the distribution of COVID-19 transmission in the U.S. and the association between BMI and structural brain connectivity of healthy individuals are also investigated.
Paper Structure (27 sections, 4 theorems, 22 equations, 4 figures, 8 tables, 2 algorithms)

This paper contains 27 sections, 4 theorems, 22 equations, 4 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Embedding metric spaces into Euclidean space
  3. Ensemble estimates of the central space
  4. General background to the ensemble technique
  5. Surrogate-assisted OLS and SIR estimators of ${\mathcal{S}}_{Y|{\bf X}}$
  6. Metric choice and estimation algorithm
  7. Metric choice for different response spaces
  8. Euclidean responses
  9. Distribution as response
  10. Network as response
  11. Functional responses
  12. Estimation algorithms
  13. Asymptotic properties of the surrogate-assisted estimator
  14. Simulation Experiments
  15. Euclidean response
  16. ...and 12 more sections

Key Result

Theorem 1

Let $\epsilon \in (0,1)$ and for any integer $n$, let $k$ be a positive integer such that $k \geq C\epsilon^{-2}\log n$. Then for any set of $n$ points $\{{\bf a}_1,\ldots,{\bf a}_n\} \in \mathbb{R}^p$, there is a mapping $f:\mathbb{R}^p \to \mathbb{R}^k$ such that

Figures (4)

  • Figure 1: Distributions of COVID-19 case transmission rate across counties in the State of Texas between 08/1/2021 and 02/21/2022.
  • Figure 2: Scatter plot of some summary statistics of the distributions of COVID-19 transmission rates vs the estimated sufficient predictor based on sa-SIR estimator.
  • Figure 3: Brain connectivity networks of the first four subjects in the data set. Node size is proportional to the degree of the node.
  • Figure 4: Scatter plot of some network properties vs BMI are provided. The blue line represents the loess curve.

Theorems & Definitions (12)

  • Definition 1: Metric
  • Definition 2: Isometric embedding
  • Definition 3: $\epsilon$-almost isometry
  • Theorem 1: Johnson–Lindenstrauss Lemma
  • Proposition 1
  • Definition 4
  • Proposition 2
  • Theorem 2
  • proof : Proof of Theorem \ref{['JL']}
  • proof : Proof of Proposition \ref{['surr_sdr']}
  • ...and 2 more