Tensor Kernel Recovery for Discrete Spatio-Temporal Hawkes Processes

Heejune Sheen, Xiaonan Zhu, Yao Xie

Abstract: We introduce a new discrete spatio-temporal Hawkes process model by formulating the general influence of the Hawkes process as a tensor kernel. Based on the low-rank structure assumption of the tensor kernel, we cast the estimation of the tensor kernel as a convex optimization problem using the Fourier transformed nuclear norm. We provide theoretical performance guarantees for our approach and present an algorithm to solve the optimization problem. In particular, our upper bound of squared estimation error has the convergence rate of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(lnK/\sqrt{K})$</tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K$</tex-math></inline-formula> is the number of samples in the time horizon. The efficiency of our estimation is demonstrated with numerical simulations on synthetic data and the analysis of real-world data from Atlanta burglary incidents.