Stationary Geometric Graphical Model Selection
This addresses the problem of efficient network inference in data-scarce scenarios for applications involving spatially embedded networks, representing a novel theoretical advancement rather than an incremental improvement.
The paper tackles model selection in Gaussian Markov fields under sample-deficient conditions by introducing spatially stationary distributions for geometric graphs, which reduces sample complexity compared to abstract graphs. They provide tight information-theoretic bounds showing that a finite number of samples suffices for consistent recovery and develop an efficient reconstruction technique.
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.