Nonlinear sufficient dimension reduction for distribution-on-distribution regression.

Area of Science:

• Statistics
• Machine Learning
• Data Science

Background:

• Sufficient dimension reduction (SDR) is crucial for analyzing high-dimensional data.
• Existing SDR methods often struggle with distributional predictor and response variables.
• Modeling data as members of a metric space offers a flexible framework.

Purpose of the Study:

• To develop a new nonlinear sufficient dimension reduction (SDR) method for distributional data.
• To address challenges posed by predictor and response variables residing in metric spaces.
• To leverage universal kernels for characterizing conditional independence in SDR.

Main Methods:

• Constructing cc-universal kernels on metric spaces to form reproducing kernel Hilbert spaces.
• Utilizing Wasserstein distance for univariate distributional data.
• Employing sliced Wasserstein distance for multivariate distributional data, balancing topological properties and computational efficiency.

Main Results:

• The proposed method demonstrates superior performance compared to competing approaches on synthetic datasets.
• The cc-universal kernels enable characterization of conditional independence essential for SDR.
• The use of sliced Wasserstein distance provides computational advantages for multivariate data.

Conclusions:

• The novel SDR approach effectively handles nonlinear relationships in distributional data.
• The method offers a robust framework for analyzing complex datasets, including fertility, mortality, and temperature data.
• This work advances SDR techniques for distributional data analysis in statistics and machine learning.