Differential calculi on finite groups

A brief review of bicovariant differential calculi on finite groups is given, with some new developments on diffeomorphisms and integration. We illustrate the general theory with the example of the nonabelian finite group S_3.Comment: LaTeX, 16 pages, 1 figur

Gravity on Finite Groups

Gravity theories are constructed on finite groups G. A self-consistent review of the differential calculi on finite G is given, with some new developments. The example of a bicovariant differential calculus on the nonabelian finite group S_3 is treated in detail, and used to build a gravity-like field theory on S_3.Comment: LaTeX, 26 pages, 1 figure. Corrected misprints and formula giving exterior product of n 1-forms. Added note on topological actio

Cycle-based Cluster Variational Method for Direct and Inverse Inference

We elaborate on the idea that loop corrections to belief propagation could be dealt with in a systematic way on pairwise Markov random fields, by using the elements of a cycle basis to define region in a generalized belief propagation setting. The region graph is specified in such a way as to avoid dual loops as much as possible, by discarding redundant Lagrange multipliers, in order to facilitate the convergence, while avoiding instabilities associated to minimal factor graph construction. We end up with a two-level algorithm, where a belief propagation algorithm is run alternatively at the level of each cycle and at the inter-region level. The inverse problem of finding the couplings of a Markov random field from empirical covariances can be addressed region wise. It turns out that this can be done efficiently in particular in the Ising context, where fixed point equations can be derived along with a one-parameter log likelihood function to minimize. Numerical experiments confirm the effectiveness of these considerations both for the direct and inverse MRF inference.Comment: 47 pages, 16 figure

On Scaling Limits of Power Law Shot-noise Fields

This article studies the scaling limit of a class of shot-noise fields defined on an independently marked stationary Poisson point process and with a power law response function. Under appropriate conditions, it is shown that the shot-noise field can be scaled suitably to have a $\alpha$-stable limit, intensity of the underlying point process goes to infinity. It is also shown that the finite dimensional distributions of the limiting random field have i.i.d. stable random components. We hence propose to call this limte the $\alpha$- stable white noise field. Analogous results are also obtained for the extremal shot-noise field which converges to a Fr\'{e}chet white noise field. Finally, these results are applied to the analysis of wireless networks.Comment: 17 pages, Typos are correcte

Exploring Two Novel Features for EEG-based Brain-Computer Interfaces: Multifractal Cumulants and Predictive Complexity

In this paper, we introduce two new features for the design of electroencephalography (EEG) based Brain-Computer Interfaces (BCI): one feature based on multifractal cumulants, and one feature based on the predictive complexity of the EEG time series. The multifractal cumulants feature measures the signal regularity, while the predictive complexity measures the difficulty to predict the future of the signal based on its past, hence a degree of how complex it is. We have conducted an evaluation of the performance of these two novel features on EEG data corresponding to motor-imagery. We also compared them to the most successful features used in the BCI field, namely the Band-Power features. We evaluated these three kinds of features and their combinations on EEG signals from 13 subjects. Results obtained show that our novel features can lead to BCI designs with improved classification performance, notably when using and combining the three kinds of feature (band-power, multifractal cumulants, predictive complexity) together.Comment: Updated with more subjects. Separated out the band-power comparisons in a companion article after reviewer feedback. Source code and companion article are available at http://nicolas.brodu.numerimoire.net/en/recherche/publication