Matthew Repasky

Zheng Dong, Matthew Repasky, Xiuyuan Cheng et al.

Point process models are widely used for continuous asynchronous event data, where each data point includes time and additional information called "marks", which can be locations, nodes, or event types. This paper presents a novel point process model for discrete event data over graphs, where the event interaction occurs within a latent graph structure. Our model builds upon Hawkes's classic influence kernel-based formulation in the original self-exciting point processes work to capture the influence of historical events on future events' occurrence. The key idea is to represent the influence kernel by Graph Neural Networks (GNN) to capture the underlying graph structure while harvesting the strong representation power of GNNs. Compared with prior works focusing on directly modeling the conditional intensity function using neural networks, our kernel presentation herds the repeated event influence patterns more effectively by combining statistical and deep models, achieving better model estimation/learning efficiency and superior predictive performance. Our work significantly extends the existing deep spatio-temporal kernel for point process data, which is inapplicable to our setting due to the fundamental difference in the nature of the observation space being Euclidean rather than a graph. We present comprehensive experiments on synthetic and real-world data to show the superior performance of the proposed approach against the state-of-the-art in predicting future events and uncovering the relational structure among data.

Matthew Repasky, Xiuyuan Cheng, Yao Xie

Learning to differentiate model distributions from observed data is a fundamental problem in statistics and machine learning, and high-dimensional data remains a challenging setting for such problems. Metrics that quantify the disparity in probability distributions, such as the Stein discrepancy, play an important role in high-dimensional statistical testing. In this paper, we investigate the role of $L^2$ regularization in training a neural network Stein critic so as to distinguish between data sampled from an unknown probability distribution and a nominal model distribution. Making a connection to the Neural Tangent Kernel (NTK) theory, we develop a novel staging procedure for the weight of regularization over training time, which leverages the advantages of highly-regularized training at early times. Theoretically, we prove the approximation of the training dynamic by the kernel optimization, namely the ``lazy training'', when the $L^2$ regularization weight is large, and training on $n$ samples converge at a rate of ${O}(n^{-1/2})$ up to a log factor. The result guarantees learning the optimal critic assuming sufficient alignment with the leading eigen-modes of the zero-time NTK. The benefit of the staged $L^2$ regularization is demonstrated on simulated high dimensional data and an application to evaluating generative models of image data.