François G. Meyer

Nicholas J. Cooper, François G. Meyer, Michael L. Roberts et al.

We introduce a unified theoretical framework for the rigorous analysis and systematic construction of deep neural networks (DNNs). This framework addresses a gap in existing theory by explicitly modeling the structure of tensor operations -- lower level information that is often abstracted. Our framework enables two novel objectives: (1) analysis of the evolution of architectural complexity over deep learning history, and (2) automatic construction of novel architectures based on new types of tensor operations. Our study of DNNs introduced over the past 40 years reveals a connection between groundbreaking architectures and increases in different types of architectural complexity. Moreover, we identify several large classes of higher complexity architectures that have not yet been explored. We then collect a dataset of 3,000+ higher complexity architectures, which we publicly release at: https://github.com/combinatoriallabs/ArchitecturalComplexity.

Daniel Ferguson, François G. Meyer

For graph-valued data sampled iid from a distribution $μ$, the sample moments are computed with respect to a choice of metric. In this work, we equip the set of graphs with the pseudo-metric defined by the $\ell_2$ norm between the eigenvalues of the respective adjacency matrices. We use this pseudo metric and the respective sample moments of a graph valued data set to infer the parameters of a distribution $\hatμ$ and interpret this distribution as an approximation of $μ$. We verify experimentally that complex distributions $μ$ can be approximated well taking this approach.

Adam Sanchez, François G. Meyer

This work addresses the rising demand for novel tools in statistical and machine learning for "graph-valued random variables" by proposing a fast algorithm to compute the sample Frechet mean, which replaces the concept of sample mean for graphs (or networks). We use convolutional neural networks to learn the morphology of the graphs in a set of graphs. Our experiments on several ensembles of random graphs demonstrate that our method can reliably recover the sample Frechet mean.

François G. Meyer

The notion of barycentre graph is of crucial importance for machine learning algorithms that process graph-valued data. The barycentre graph is a "summary graph" that captures the mean topology and connectivity structure of a training dataset of graphs. The construction of a barycentre requires the definition of a metric to quantify distances between pairs of graphs. In this work, we use a multiscale spectral distance that is defined using the eigenvalues of the normalized graph Laplacian. The eigenvalues -- but not the eigenvectors -- of the normalized Laplacian of the barycentre graph can be determined from the optimization problem that defines the barycentre. In this work, we propose a structural constraint on the eigenvectors of the normalized graph Laplacian of the barycentre graph that guarantees that the barycentre inherits the topological structure of the graphs in the sample dataset. The eigenvectors can be computed using an algorithm that explores the large library of Soules bases. When the graphs are random realizations of a balanced stochastic block model, then our algorithm returns a barycentre that converges asymptotically (in the limit of large graph size) almost-surely to the population mean of the graphs. We perform Monte Carlo simulations to validate the theoretical properties of the estimator; we conduct experiments on real-life graphs that suggest that our approach works beyond the controlled environment of stochastic block models.

Daniel Ferguson, François G. Meyer

To characterize the location (mean, median) of a set of graphs, one needs a notion of centrality that is adapted to metric spaces, since graph sets are not Euclidean spaces. A standard approach is to consider the Fréchet mean. In this work, we equip a set of graph with the pseudometric defined by the $\ell_2$ norm between the eigenvalues of their respective adjacency matrix . Unlike the edit distance, this pseudometric reveals structural changes at multiple scales, and is well adapted to studying various statistical problems on sets of graphs. We describe an algorithm to compute an approximation to the Fréchet mean of a set of undirected unweighted graphs with a fixed size.

François G. Meyer, Alexander M. Benison, Zachariah Smith et al.

We describe here the recent results of a multidisciplinary effort to design a biomarker that can actively and continuously decode the progressive changes in neuronal organization leading to epilepsy, a process known as epileptogenesis. Using an animal model of acquired epilepsy, wechronically record hippocampal evoked potentials elicited by an auditory stimulus. Using a set of reduced coordinates, our algorithm can identify universal smooth low-dimensional configurations of the auditory evoked potentials that correspond to distinct stages of epileptogenesis. We use a hidden Markov model to learn the dynamics of the evoked potential, as it evolves along these smooth low-dimensional subsets. We provide experimental evidence that the biomarker is able to exploit subtle changes in the evoked potential to reliably decode the stage of epileptogenesis and predict whether an animal will eventually recover from the injury, or develop spontaneous seizures.