Stationary Geometric Graphical Model Selection
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We consider the problem of model selection in Gaussian Markov fields in the sample deficient scenario. In many cases, the underlying networks are embedded into Euclidean spaces which induces significant structure on them. Using this natural spatial structure, we introduce the notion of spatially stationary distributions over geometric graphs directly generalizing the notion of stationary time series to the multidimensional setup 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 the 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.
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