Manifold Regression

2026-10-08

AI Summary

This study addresses the limitation of traditional regression models in handling non-one-to-one mappings by proposing a multi-valued regression framework based on latent manifolds. Methodologically, it transcends the single-valued assumption through parametric latent manifold modeling, subsuming classical regression as a unified special case. By integrating regularized contour optimization with a robust solver, the approach achieves theoretically grounded, slice-wise set-valued prediction. The primary contribution lies in establishing a multi-valued prediction mechanism endowed with rigorous theoretical guarantees. Experimental validation demonstrates that the proposed method delivers stable and superior predictive performance in complex non-one-to-one scenarios.

Abstract

Conventional statistical modeling typically assumes a one-to-one mapping between input and output variables, where one-to-one is used in the predictive sense that a specified input results in a unique output. Many scientific and engineering systems, however, exhibit non-one-to-one(noto) input-output relations. In a noto relation, the same input may correspond to multiple admissible outputs, so the conventional statistical models may not be appropriate. This paper develops manifold regression, a parametric modeling framework for such noto input-output relations, which represents the underlying input-output relation through a latent manifold. The manifold is specified upto an unknown finite dimensional parameter and we estimate the unknown parameters from ordinary input-output observations by regularized profile optimization with a robust solver. Once the manifold has been learned, prediction is obtained by slicing the estimated manifold along a specified coordinate value. Because a slice may contain multiple latent roots, the resulting manifold prediction is naturally set-valued. This formulation extends predictive learning beyond conventional one-to-one statistical models while containing classical regression and inverse prediction as special cases. Theoretical results are established to justify the manifold regression model and its sliced prediction sets; and case studies are used to demonstrate stable performance across representative noto cases.

Problem

• Manifold Regression
• Non-one-to-one mapping
• Set-valued prediction
• Statistical modeling
• Latent manifold

Innovation

• Manifold Regression
• Non-one-to-one Relations
• Set-valued Prediction
• Latent Manifold
• Regularized Profile Optimization