Sparse space-time models: Concentration Inequalities and Lasso

TL;DR

This paper introduces a new stochastic model for infinite neuronal networks inspired by Kalikow-type decompositions, providing sharp oracle inequalities for Lasso and restricted eigenvalue properties, even with partial observations.

Contribution

It presents a novel sparse space-time model with theoretical guarantees for Lasso estimation in complex neuronal networks.

Findings

Sharp oracle inequalities for Lasso methods.

Restricted eigenvalue properties for the Gram matrix.

Results hold under partial network observation.

Abstract

Inspired by Kalikow-type decompositions, we introduce a new stochastic model of infinite neuronal networks, for which we establish sharp oracle inequalities for Lasso methods and restricted eigenvalue properties for the associated Gram matrix with high probability. These results hold even if the network is only partially observed. The main argument rely on the fact that concentration inequalities can easily be derived whenever the transition probabilities of the underlying process admit a sparse space-time representation.