Kernel-Based Reconstruction of Space-Time Functions on Dynamic Graphs

Graph-based methods pervade the inference toolkits of numerous disciplines including sociology, biology, neuroscience, physics, chemistry, and engineering. A challenging problem encountered in this context pertains to determining the attributes of a set of vertices given those of another subset at possibly diffe-rent time instants. Leveraging spatiotemporal dynamics can drastically reduce the number of observed vertices, and hence the sampling cost. Alleviating the limited flexibility of the existing approaches, the present paper broadens the kernel-based graph function estimation framework to reconstruct time-evolving functions over possibly time-evolving topologies. This approach inherits the versatility and generality of kernel-based methods, for which no knowledge on distributions or second-order statistics is required. Systematic guidelines are provided to construct two families of space-time kernels with complementary strengths: the first facilitates judicious control of regularization on a space-time frequency plane, whereas the second accommodates time-varying topologies. Batch and online estimators are also put forth. The latter comprise a novel kernel Kalman filter, developed to reconstruct space-time functions at affordable computational cost. Numerical tests with real datasets corroborate the merits of the proposed methods relative to competing alternatives.