B\'ezier Curve Gaussian Processes

TL;DR

This paper introduces a Bayesian approach to probabilistic Bézier curve models for sequential data, bridging the gap between computational efficiency and expressiveness, demonstrated through human trajectory prediction.

Contribution

It presents a novel method to perform full Bayesian inference on Bézier curve Gaussian processes, enhancing expressiveness over traditional Mixture Density Networks.

Findings

Enables Bayesian inference on Bézier curve models

Derives mean and kernel functions for different data modalities

Improves human trajectory prediction accuracy

Abstract

Probabilistic models for sequential data are the basis for a variety of applications concerned with processing timely ordered information. The predominant approach in this domain is given by recurrent neural networks, implementing either an approximate Bayesian approach (e.g. Variational Autoencoders or Generative Adversarial Networks) or a regression-based approach, i.e. variations of Mixture Density networks (MDN). In this paper, we focus on the $\mathcal{N}$-MDN variant, which parameterizes (mixtures of) probabilistic B\'ezier curves ($\mathcal{N}$-Curves) for modeling stochastic processes. While in favor in terms of computational cost and stability, MDNs generally fall behind approximate Bayesian approaches in terms of expressiveness. Towards this end, we present an approach for closing this gap by enabling full Bayesian inference on top of $\mathcal{N}$-MDNs. For this, we show that $\mathcal{N}$-Curves are a special case of Gaussian processes (denoted as $\mathcal{N}$-GP) and then derive corresponding mean and kernel functions for different modalities. Following this, we propose the use of the $\mathcal{N}$-MDN as a data-dependent generator for $\mathcal{N}$-GP prior distributions. We show the advantages granted by this combined model in an application context, using human trajectory prediction as an example.