Kernel-based Reconstruction of Space-time Functions on Dynamic Graphs

Abstract

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 different time instants. Leveraging spatiotemporal dynamics can drastically reduce the number of observed vertices, and hence the cost of sampling. Alleviating the limited flexibility of existing approaches, the present paper broadens the existing kernel-based graph function reconstruction framework to accommodate 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 can afford time-varying topologies. Batch and online estimators are also put forth, and a novel kernel Kalman filter is developed to obtain these estimates at affordable computational cost. Numerical tests with real data sets corroborate the merits of the proposed methods relative to competing alternatives.

Inference of spatiotemporal processes over graphs via kernel kriged Kalman filtering

“…A time-invariant graph was constructed as in [10], based on geographical distances. The value f n [t] represents the temperature recorded at the n-th station and t-th sample.…”

“…However, they rely on the bandlimited model, whose effectiveness in capturing the dynamics of real-world graph functions may not hold. A kernel-based Kalman filter that captures muliple forms of spatiotemporal dynamics through space-time kernels was explored in [10]. But it mainly relies on smoothness and does not explicitly account for the underlying dynamics.…”

Distributed recursive least squares strategies for adaptive reconstruction of graph signals

“…Based on the developed sampling theory, several reconstruction methods were proposed, either iterative as in [13], [15], or batch, as in [6], [12]. On the other hand, [16]- [18] propose adaptive strategies for online graph signal reconstruction and learning. Specifically, [16] proposed an LMS estimation strategy enabling adaptive learning and tracking from a limited number of smartly sampled observations, which…”

Optimal sampling strategies for adaptive learning of graph signals

“…Then, several reconstruction methods have been proposed, either iterative as in [13], [14], or batch, as in [8], [10], [15]. Recently, adaptive strategies for online reconstruction and learning of graph signals were also proposed in [16]- [18], and paved the way to the development of novel adaptive GSP tools. In particular, reference [16] proposed an LMS estimation strategy for adaptive reconstruction of graph signals from a subset of samples smartly collected over the graph.…”

“…The method was then extended to the distributed setting in [17]. Finally, in [18], the authors proposed a kernel-based reconstruction framework to handle functions evolving over possibly time-varying topologies, leveraging spatio-temporal dynamics of the observed graph signals.…”