simulating determinantal processes

In the plane to Atlanta, I happened to read a paper called Efficient simulation of the Ginibre point process by Laurent Decreusefond, Ian Flint, and Anaïs Vergne (from Telecom Paristech). “Happened to” as it was a conjunction of getting tipped by my new Dauphine colleague (and fellow blogger!) Djalil Chaffaï about the paper, having downloaded it prior to departure, and being stuck in a plane (after watching the only Chinese [somewhat] fantasy movie onboard, Saving General Yang).

This is mostly a mathematics paper. While indeed a large chunk of it is concerned with the rigorous definition of this point process in an abstract space, the last part is about simulating such processes. They are called determinantal (and not detrimental as I was tempted to interpret on my first read!) because the density of an n-set (x₁, x₂,…,x_n) is given by a kind of generalised Vandermonde determinant

where T is defined in terms of an orthonormal family,

(The number n of points can be simulated via an a.s. finite Bernoulli process.) Because of this representation, the sequence of conditional densities for the x_i‘s (i.e. x₁, x₂ given x₁, etc.) can be found in closed form. In the special case of the Ginibre process, the ψ_i‘s are of the form

and the process cannot be simulated for it has infinite mass, hence an a.s. infinite number of points. Somehow surprisingly (as I thought this was the point of the paper), the authors then switch to a truncated version of the process that always has a fixed number N of points. And whose density has the closed form

It has an interestingly repulsive quality in that points cannot get close to one another. (It reminded me of the pinball sampler proposed by Kerrie Mengersen and myself at one of the Valencia meetings and not pursued since.) The conclusion (of this section) is anticlimactic, though,  in that it is known that this density also corresponds to the distribution of the eigenvalues of an Hermitian matrix with standardized complex Gaussian entries. The authors mentions that the fact that the support is the whole complex space Cⁿ is a difficulty, although I do not see why.

The following sections of the paper move to the Ginibre process restricted to a compact and then to the truncated Ginibre process restricted to a compact, for which the authors develop corresponding simulation algorithms. There is however a drag in that the sequence of conditionals, while available in closed-form, cannot be simulated efficiently but rely on a uniform accept-reject instead. While I am certainly missing most of the points in the paper, I wonder if a Gibbs sampler would not be an interesting alternative given that the full (last) conditional is a Gaussian density…

non-parametric Bayes at Telecom ParisTech

On Thursday, September 6, Telecom ParisTech holds a conference on “Méthodes bayésiennes non paramétriques pour le Traitement du Signal et des Images“. Attendance is free. It will start at 10am with a tutorial by Michael Jordan who is visiting Paris this academic year (great!).

slideshare m’a tuer…

This afternoon, I completely botched my talk at the workshop “Méthodes de Monte Carlo pour les problèmes inverses bayésiens en traitement des signaux et des images” thanks to a weird mishap on slideshare. As Og’s readers are aware,  I am a big fan of slideshare as a public depository for my slides. However, when I uploaded my talk this morning, “something” happens with the pdf file in that, while I was seeing a regular talk on my own computer (using acroread Paris.pdf), the output of the file on slideshare was missing integrals, sum and products! This was most embarrassing for me and most  inconvenient for the audience as I had to explain over and over the meaning of the missing symbols… I could have stopped the talk and gathered the pdf file from my laptop, of course, but this was the last session of a busy day, I was not sure the original was all-right ,and I had no USB key nor power plug with me… The audience was much sympathetic than I would have been myself. Apologies to all! The most bizarre thing is that I have been trying to reproduce the phenomenon since then, but to no avail. Every version I have downloaded from this afternoon onwards contains the right symbols (on my screen). Maybe it was due to the resolution of the local computer I was using for my presentation as it could not produce a full screen output… Anyway, an embarrassment I could have done without! (The title of this post is inspired from a famous crime story in southern France where the victim supposedly wrote “Omar m’a tuer” on a wall with her blood, committing a very unlikely grammatical mistake… This erroneous sentence became almost immediately part of the urban culture. I once got a “Robert m’a tuer…” as a course evaluation the only year I gave a course on classical testing!!! A movie on the crime came out recently.)

Postdoc position at Telecom Paris

Another announcement I got today, for an interesting postdoc with Olivier Cappé at Telecom, covering potentially Bayesian applications:

A one year post-doctoral position is offered at the Laboratoire Traitement et Communication de l’Information (LTCI), the joint laboratory of CNRS and Telecom ParisTech (aka ENST), physically located in Telecom ParisTech‘s buildings in downtown Paris.

The position is within the project MGA (Graphical Models and Applications), funded by the French National Research Agency (ANR). The MGA project is a joint research project with teams from INRIA (Francis Bach), Mines ParisTech (Jean-Philippe Vert) and Ecole des Ponts ParisTech (Jean-Yves Audibert). The MGA projects deals with graphical models (in a broad sense) and applications in bioinformatics, computer vision and natural language processing. The MGA team is, among other things, organizing a regular seminar on machine learning that gathers researchers and students from Paris area in mathematics, computer science, and various application domains. Within LTCI, the postdoc will be in the STA (Statistics and Applications) Group and supervised by Olivier Cappé.

Candidates should have a PhD in the areas of machine learning, computational statistics or signal processing. Familiarity with (at least one of) kernel methods, graphical models, Bayesian computation, inference in latent variable models will be appreciated.

The postdoctoral researcher will be employed by the CNRS with a net salary of about 2000 EUR per month. The position is for one year, starting January 2010. Candidates should send a detailed CV, including a list of publications to Olivier Cappé (cappe [à] telecom-paristech.fr)