Manifold Regression—Design and Analysis

Abstract

Conventional statistical modeling typically assumes a one-to-one mapping between input and output variables. 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 talk proposes some recent developments on manifold regression for such NOTO input-output relations, which represents the underlying input-output relation through a latent manifold. Both parametric and nonparametric approaches are discussed. If time permits,some design issues will also be discussed.For parametric approach, the manifold is specified upto an unknown finite dimensional parameter and then 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. For nonparametric approach, we first find the latent ordering, encode it by equi-spaced coordinates for regularized nonparametric fitting, and then use an anchored augmented Lagrangian step to refine the latent coordinates locally toward the input equalities. We show that the exact latent coordinates are not identifiable under monotone reparameterization, whereas their ordering is invariant up to reversal. Theoretical results are established to justify the manifold regression model and its sliced prediction sets. Case studies are used to demonstrate a stable performance across representative NOTO cases.

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