Inverting Non-Injective Functions with Twin Neural Network Regression
This work addresses the challenge of deterministic inversion for non-injective functions, which is incremental as it builds on existing inversion methods by introducing a local anchoring technique to handle multi-valued mappings in domains like robotics.
The paper tackles the problem of inverting non-injective functions, which lack global invertibility, by proposing Twin Neural Network Regression to predict local inverse corrections around anchor points, enabling deterministic selection of valid branches; it demonstrates the method on mathematical and data-driven problems, such as robot arm inverse kinematics, achieving consistent inversion without probabilistic approaches.
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.