#### Project: GenGP

##### November 25, 2018

GenGP is a project funded by ANR (the French National Agency for Research). We are investigating numerical methods for Gaussian Processes, with applications to Determinantal Point Processes.

## Gaussian processes

Gaussian processes GPs are a convenient way of defining priors over unknown functions: in an inference problem, they allow you to state that a function is smooth, without assuming a specific form (in other words, they allow non-parametric inference in a Bayesian setting).

The figure below shows samples from a GP, interpreted as a prior distribution: we have a an unknown function \(f(x)\) and all we wish to state is that \(f\) is smooth. The next panel shows a posterior distribution: we have a few noisy measurements of \(f\) (\(y_1 \ldots y_n\), and the black lines are samples from the posterior distribution, \(p(f|y_1 \ldots y_n)\). The beauty of GPs is that the posterior distribution is again a GP, but one constrained to lie close to the measured values.

For more on GPs see the wonderful book by Rasmussen & Williams, Gaussian Processes for Machine Learning.

## Challenges

GPs are notorious for scaling badly when the number of observations increases, with a cost in \(\mathcal{O}(n^{3})\). There are dozens of approximations attempting to reduce cost, and one of the most successful is based on a sparse approximation of the inverse covariance matrix (thatâ€™s the method implemented in INLA, and described in Lindgren et al. 2011). The theoretical underpinnings are somewhat complicated and rely crucially on seeing the GP as the solution of a stochastic Partial Differential Equations.

Unfortunately, this limits the applicability of the method to low-dimensional manifolds, but GPs are much more generic: as soon as you can define a covariance function over a space, you can define a GP over a space. Thus, one can have GPs over spaces of graphs, of strings, etc.

The goal of the GenGP project is to generalise the sparse-inverse approach to GPs.

## Applications

So far we have worked a lot on Determinantal Point Processes, which are very tightly connected to GPs.

## People

Ronald Phlypo Konstantin Usevich

On DPPs I work closely with Nicolas Tremblay and Pierre-Olivier Amblard.