# B'ezier Curve Gaussian Processes

arXiv: https://arxiv.org/abs/2205.01754 · HTML: https://mlsanity.com/paper/2205.01754/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 38 references that are in the map.
- First posted on arXiv 3 May 2022.

## Builds on (most cited first, 11 of 11)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [Generating Sentences from a Continuous Space](https://mlsanity.com/paper/1511.06349/) (2015)
- [Generating Sequences With Recurrent Neural Networks](https://mlsanity.com/paper/1308.0850/) (2013)
- [SeqGAN: Sequence Generative Adversarial Nets with Policy Gradient](https://mlsanity.com/paper/1609.05473/) (2016)
- [Human Trajectory Forecasting in Crowds: A Deep Learning Perspective](https://mlsanity.com/paper/2007.03639/) (2020)
- [Distributed Gaussian Processes](https://mlsanity.com/paper/1502.02843/) (2015)
- [Accurate and Diverse Sampling of Sequences based on a "Best of Many" Sample Objective](https://mlsanity.com/paper/1806.07772/) (2018)
- [Multivariate Gaussian and Student-t Process Regression for Multi-output Prediction](https://mlsanity.com/paper/1703.04455/) (2017)
- [Recurrent Gaussian Processes](https://mlsanity.com/paper/1511.06644/) (2015)

## Built on by (most cited first, 3 of 3)

- [EigenTrajectory: Low-Rank Descriptors for Multi-Modal Trajectory Forecasting](https://mlsanity.com/paper/2307.09306/) (2023)
- [B'ezier Gaussian Processes for Tall and Wide Data](https://mlsanity.com/paper/2209.00343/) (2022)
- [Generating Synthetic Ground Truth Distributions for Multi-step Trajectory Prediction using Probabilistic Composite B'ezier Curves](https://mlsanity.com/paper/2404.04397/) (2024)

## Abstract (arXiv)

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 N-MDN variant, which parameterizes (mixtures of) probabilistic B'ezier curves (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 N-MDNs. For this, we show that N-Curves are a special case of Gaussian processes (denoted as N-GP) and then derive corresponding mean and kernel functions for different modalities. Following this, we propose the use of the N-MDN as a data-dependent generator for N-GP prior distributions. We show the advantages granted by this combined model in an application context, using human trajectory prediction as an example.