Level set learning with pseudo-reversible neural networks for nonlinear dimension reduction in function approximation

TL;DR

This paper introduces DRiLLS, a novel dimension reduction method using pseudo-reversible neural networks for efficient high-dimensional function approximation, outperforming existing methods especially with interior critical points.

Contribution

The paper proposes a new DRiLLS method combining PRNN and local regression, relaxing invertibility constraints and improving approximation accuracy over NLL and Active Subspace methods.

Findings

DRiLLS outperforms NLL and Active Subspace methods in experiments.

PRNN effectively transforms high-dimensional inputs to low-dimensional active variables.

Local regression with Euclidean distance reduces numerical oscillations.

Abstract

Due to the curse of dimensionality and the limitation on training data, approximating high-dimensional functions is a very challenging task even for powerful deep neural networks. Inspired by the Nonlinear Level set Learning (NLL) method that uses the reversible residual network (RevNet), in this paper we propose a new method of Dimension Reduction via Learning Level Sets (DRiLLS) for function approximation. Our method contains two major components: one is the pseudo-reversible neural network (PRNN) module that effectively transforms high-dimensional input variables to low-dimensional active variables, and the other is the synthesized regression module for approximating function values based on the transformed data in the low-dimensional space. The PRNN not only relaxes the invertibility constraint of the nonlinear transformation present in the NLL method due to the use of RevNet, but also adaptively weights the influence of each sample and controls the sensitivity of the function to the learned active variables. The synthesized regression uses Euclidean distance in the input space to select neighboring samples, whose projections on the space of active variables are used to perform local least-squares polynomial fitting. This helps to resolve numerical oscillation issues present in traditional local and global regressions. Extensive experimental results demonstrate that our DRiLLS method outperforms both the NLL and Active Subspace methods, especially when the target function possesses critical points in the interior of its input domain.