Hyper Input Convex Neural Networks for Shape Constrained Learning and Optimal Transport

TL;DR

Hyper Input Convex Neural Networks (HyCNNs) are a new architecture that efficiently learns convex functions, outperforming existing methods in convex regression, interpolation, and high-dimensional optimal transport tasks.

Contribution

HyCNNs combine Maxout and input convex neural networks to create a scalable, convex neural network architecture with fewer parameters and improved performance.

Findings

HyCNNs require exponentially fewer parameters than ICNNs for quadratic approximation.

HyCNNs outperform ICNNs and MLPs in convex regression and interpolation tasks.

HyCNNs often outperform ICNN-based optimal transport methods on high-dimensional data.

Abstract

We introduce Hyper Input Convex Neural Networks (HyCNNs), a novel neural network architecture designed for learning convex functions. HyCNNs combine the principles of Maxout networks with input convex neural networks (ICNNs) to create a neural network that is always convex in the input, theoretically capable of leveraging depth, and performs reliable when trained at scale compared to ICNNs. Concretely, we prove that HyCNNs require exponentially fewer parameters than ICNNs to approximate quadratic functions up to a given precision. Throughout a series of synthetic experiments, we demonstrate that HyCNNs outperform existing ICNNs and MLPs in terms of predictive performance for convex regression and interpolation tasks. We further apply HyCNNs to learn high-dimensional optimal transport maps for synthetic examples and for single-cell RNA sequencing data, where they oftentimes outperform ICNN-based neural optimal transport methods and other baselines across a wide range of settings.