Common Substructure Learning of Multiple Graphical Gaussian Models

Learning underlying mechanisms of data generation is of great interest in the scientific and engineering fields amongst others. Finding dependency structures among variables in the data is one possible approach for the purpose, and is an important task in data mining. In this paper, we focus on learning dependency substructures shared by multiple datasets. In many scenarios, the nature of data varies due to a change in the surrounding conditions or non-stationary mechanisms over the multiple datasets. However, we can also assume that the change occurs only partially and some relations between variables remain unchanged. Moreover, we can expect that such commonness over the multiple datasets is closely related to the invariance of the underlying mechanism. For example, errors in engineering systems are usually caused by faults in the sub-systems with the other parts remaining healthy. In such situations, though anomalies are observed in sensor values, the underlying invariance of the healthy sub-systems is still captured by some steady dependency structures before and after the onset of the error. We propose a structure learning algorithm to find such invariances in the case of Graphical Gaussian Models (GGM). The proposed method is based on a block coordinate descent optimization, where subproblems can be solved efficiently by existing algorithms for

Lasso

and the

continuous quadratic knapsack problem

. We confirm the validity of our approach through numerical simulations and also in applications with real world datasets extracted from the analysis of city-cycle fuel consumption and anomaly detection in car sensors.