Learning Invariances with Generalised Input-Convex Neural Networks

TL;DR

This paper introduces a new class of neural networks that guarantee connected level sets, enabling efficient global parameterisation of invariance manifolds for data exploration and analysis.

Contribution

The paper proposes a novel generalised input-convex neural network architecture that ensures connected level sets, facilitating manifold characterisation and data exploration tasks.

Findings

Networks guarantee connected level sets

Global parameterisations can be efficiently computed

Effective in real-world applications like computational chemistry

Abstract

Considering smooth mappings from input vectors to continuous targets, our goal is to characterise subspaces of the input domain, which are invariant under such mappings. Thus, we want to characterise manifolds implicitly defined by level sets. Specifically, this characterisation should be of a global parametric form, which is especially useful for different informed data exploration tasks, such as building grid-based approximations, sampling points along the level curves, or finding trajectories on the manifold. However, global parameterisations can only exist if the level sets are connected. For this purpose, we introduce a novel and flexible class of neural networks that generalise input-convex networks. These networks represent functions that are guaranteed to have connected level sets forming smooth manifolds on the input space. We further show that global parameterisations of these level sets can be always found efficiently. Lastly, we demonstrate that our novel technique for characterising invariances is a powerful generative data exploration tool in real-world applications, such as computational chemistry.