Inverting Non-Injective Functions with Twin Neural Network Regression

TL;DR

This paper introduces Twin Neural Network Regression, a deterministic method for inverting non-injective functions by predicting local inverse corrections, effectively handling multi-valued inverses in various applications.

Contribution

It reformulates inverse learning for non-injective functions as locally invertible problems and proposes a deterministic twin neural network approach to select valid inverse branches.

Findings

Successfully applied to mathematical and data-driven problems.

Outperforms probabilistic methods in deterministic inverse prediction.

Effective in robot arm inverse kinematics and multi-solution scenarios.

Abstract

Non-injective functions are not globally invertible. However, they can often be restricted to locally injective subdomains where the inversion is well-defined. In many settings a preferred solution can be selected even when multiple valid preimages exist or input and output dimensions differ. This manuscript describes a natural reformulation of the inverse learning problem for non-injective functions as a collection of locally invertible problems. More precisely, Twin Neural Network Regression is trained to predict local inverse corrections around known anchor points. By anchoring predictions to points within the same locally invertible region, the method consistently selects a valid branch of the inverse. In contrast to current probabilistic state-of-the art inversion methods, Inverse Twin Neural Network Regression is a deterministic framework for resolving multi-valued inverse mappings. I demonstrate the approach on problems that are defined by mathematical equations or by data, including multi-solution toy problems and robot arm inverse kinematics.