Introducing Probabilistic Bézier Curves for N-Step Sequence Prediction

Abstract

Representations of sequential data are commonly based on the assumption that observed sequences are realizations of an unknown underlying stochastic process, where the learning problem includes determination of the model parameters. In this context, a model must be able to capture the multi-modal nature of the data, without blurring between single modes. This paper proposes probabilistic B'ezier curves (𝒩-Curves) as a basis for effectively modeling continuous-time stochastic processes. The model is based on Mixture Density Networks (MDN) and B'ezier curves with Gaussian random variables as control points. Key advantages of the model include the ability of generating smooth multi-mode predictions in a single inference step which reduces the need for Monte Carlo simulation. This property is in line with recent attempts to address the problem of quantifying uncertainty as a regression problem. Essential properties of the proposed approach are illustrated by several toy examples and the task of multi-step sequence prediction. As an initial proof of concept, the model performance is compared to an LSTM-MDN model and recurrent Gaussian processes on two real world use-cases, trajectory prediction and motion capture sequence prediction.