Stationary Geometric Graphical Model Selection

TL;DR

This paper introduces spatial stationarity for geometric graphs in Gaussian Markov fields, significantly reducing sample complexity for model selection and providing efficient reconstruction methods.

Contribution

It generalizes stationarity to geometric graphs, deriving tight bounds on sample complexity and proposing an efficient reconstruction technique.

Findings

Spatial stationarity reduces sample complexity in geometric graph model selection.

Finite samples suffice for consistent graph recovery in bounded-length geometric graphs.

An efficient method for reliable graph reconstruction with limited measurements.

Abstract

We consider the problem of model selection in Gaussian Markov fields in the sample deficient scenario. In many practically important cases, the underlying networks are embedded into Euclidean spaces. Using the natural geometric structure, we introduce the notion of spatially stationary distributions over geometric graphs. This directly generalizes the notion of stationary time series to the multidimensional setting lacking time axis. We show that the idea of spatial stationarity leads to a dramatic decrease in the sample complexity of the model selection compared to abstract graphs with the same level of sparsity. For geometric graphs on randomly spread vertices and edges of bounded length, we develop tight information-theoretic bounds on sample complexity and show that a finite number of independent samples is sufficient for a consistent recovery. Finally, we develop an efficient technique capable of reliably and consistently reconstructing graphs with a bounded number of measurements.