Overview
Gaussian process dynamical models (state-space models) builds upon a long line of work combining Gaussian processes (GP) with latent variable models for unsupervised learning tasks. Specifically, we narrow our focus on modelling high-dimensional sequential data ubiquitous in nature. The dynamics in observed space are captured by a smoothly evolving latent variable indexed by time and governed by a latent Gaussian process prior. The idea behind the project is to develop a scalable algorithm for sequential data which does not rely on holding the complete sequence in memory but can process the time-series or sequence in chunks. We also want to amortise the model with a suitable encoder like a recurrent neural network, LSTM or an Attention-based transformer.
The model is basically an extension of the model proposed here: https://gregorygundersen.com/blog/2020/07/24/gpdm/
Required reading:
[1] Gaussian process dynamical model: https://gregorygundersen.com/blog/2020/07/24/gpdm/
[2] Variational GPDM: https://proceedings.neurips.cc/paper/2011/file/af4732711661056eadbf798ba191272a-Paper.pdf
[3] Variational GPSSM: https://proceedings.neurips.cc/paper/2014/file/139f0874f2ded2e41b0393c4ac5644f7-Paper.pdf
[4] Recurrent Gaussian processes with SVI: https://arxiv.org/abs/1511.06644
[5] GPVAE for interpretable latent dynamics: http://proceedings.mlr.press/v118/pearce20a/pearce20a.pdf
FAQs
- What will I learn in this Project?
The student will have to learn the basic comparitive models which work on sequential data processing in an unsupervised context - like Hidden markov models, GP-VAEs and RNNs. They will also learn the fundamentals of learning with a Gaussian process as well as coding it up within one of the popular python frameworks like gpytorch. Since this is methodological work, the student will have to perform experiments on a range of datasets so they will learn how to write code and conduct experiments at scale.
- What is the objective of the project?
A Gaussian process dynamical model (GPDM) can be viewed as a Gaussian process latent variable model (GPLVM) with the latent variable evolving according to its own Gaussian process. Put more simply, imagine a latent variable evolves according to some smooth, nonlinear dynamics, and that the mapping from latent- to observation-space is also a smooth, nonlinear map. For example, imagine a mouse is moving in a maze, but we only record the firing rates of its hippocampal place cells. The latent variable is the mouse position, which we know is a relatively smooth trajectory in 3D space, and the observations, neuron firing rates, are a nonlinear function of this latent variable. Can we infer the position of the mouse given just its firing rates? This is the type of question the project attempts to answer. The goal of this project is to work through this model in detail.
- How does this fit into the bigger picture?
Dynamical models are essentially time-indexed models where their properties evolve over time. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manifold. The evolution rule of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic, that is, for a given time interval only one future state follows from the current state. However, some systems are stochastic, in that random events also affect the evolution of the state variables.
While models are very precise for many processes, for some of the most challenging applications of dynamical systems (such as climate dynamics, brain dynamics, biological systems or the financial markets), the development of such models is notably difficult.
How to analyse dynamical systems on the basis of observed data rather than attempt to study models analytically? The machine learning approach is invaluable in settings where no explicit model is formulated, but measurement data is available. This is frequently the case in many systems of interest, and the development of data-driven technologies is becoming increasingly important in many applications.
