Salehkaleybar, Saber (University of Illinois at Urbana-Champaign) | Etesami, Jalal (University of Illinois at Urbana-Champaign) | Kiyavash, Negar (University of Illinois at Urbana-Champaign) | Zhang, Kun (Carnegie Mellon University)

We study the problem of learning the support of transition matrix between random processes in a Vector Autoregressive (VAR) model from samples when a subset of the processes are latent. It is well known that ignoring the effect of the latent processes may lead to very different estimates of the influences among observed processes, and we are concerned with identifying the influences among the observed processes, those between the latent ones, and those from the latent to the observed ones. We show that the support of transition matrix among the observed processes and lengths of all latent paths between any two observed processes can be identified successfully under some conditions on the VAR model. From the lengths of latent paths, we reconstruct the latent subgraph (representing the influences among the latent processes) with a minimum number of variables uniquely if its topology is a directed tree. Furthermore, we propose an algorithm that finds all possible minimal latent graphs under some conditions on the lengths of latent paths. Our results apply to both non-Gaussian and Gaussian cases, and experimental results on various synthetic and real-world datasets validate our theoretical results.

Fox, Emily B., Jordan, Michael I.

In this article we discuss some of the consequences of the mixed membership perspective on time series analysis. In its most abstract form, a mixed membership model aims to associate an individual entity with some set of attributes based on a collection of observed data. Although much of the literature on mixed membership models considers the setting in which exchangeable collections of data are associated with each member of a set of entities, it is equally natural to consider problems in which an entire time series is viewed as an entity and the goal is to characterize the time series in terms of a set of underlying dynamic attributes or "dynamic regimes". Indeed, this perspective is already present in the classical hidden Markov model, where the dynamic regimes are referred to as "states", and the collection of states realized in a sample path of the underlying process can be viewed as a mixed membership characterization of the observed time series. Our goal here is to review some of the richer modeling possibilities for time series that are provided by recent developments in the mixed membership framework.

Frigola, Roger, Lindsten, Fredrik, Schön, Thomas B., Rasmussen, Carl E.

State-space models are successfully used in many areas of science, engineering and economics to model time series and dynamical systems. We present a fully Bayesian approach to inference \emph{and learning} (i.e. state estimation and system identification) in nonlinear nonparametric state-space models. We place a Gaussian process prior over the state transition dynamics, resulting in a flexible model able to capture complex dynamical phenomena. To enable efficient inference, we marginalize over the transition dynamics function and infer directly the joint smoothing distribution using specially tailored Particle Markov Chain Monte Carlo samplers. Once a sample from the smoothing distribution is computed, the state transition predictive distribution can be formulated analytically. Our approach preserves the full nonparametric expressivity of the model and can make use of sparse Gaussian processes to greatly reduce computational complexity.

Li, Lin, Swami, Ananthram, Scaglione, Anna

We propose a probabilistic modeling framework for learning the dynamic patterns in the collective behaviors of social agents and developing profiles for different behavioral groups, using data collected from multiple information sources. The proposed model is based on a hierarchical Bayesian process, in which each observation is a finite mixture of an set of latent groups and the mixture proportions (i.e., group probabilities) are drawn randomly. Each group is associated with some distributions over a finite set of outcomes. Moreover, as time evolves, the structure of these groups also changes; we model the change in the group structure by a hidden Markov model (HMM) with a fixed transition probability. We present an efficient inference method based on tensor decompositions and the expectation-maximization (EM) algorithm for parameter estimation.

Xhonneux, Louis-Pascal A. C., Qu, Meng, Tang, Jian

This paper builds the connection between graph neural networks and traditional dynamical systems. Existing graph neural networks essentially define a discrete dynamic on node representations with multiple graph convolution layers. We propose continuous graph neural networks (CGNN), which generalise existing graph neural networks into the continuous-time dynamic setting. The key idea is how to characterise the continuous dynamics of node representations, i.e. the derivatives of node representations w.r.t. time. Inspired by existing diffusion-based methods on graphs (e.g. PageRank and epidemic models on social networks), we define the derivatives as a combination of the current node representations, the representations of neighbors, and the initial values of the nodes. We propose and analyse different possible dynamics on graphs---including each dimension of node representations (a.k.a. the feature channel) change independently or interact with each other---both with theoretical justification. The proposed continuous graph neural networks are robust to over-smoothing and hence capture the long-range dependencies between nodes. Experimental results on the task of node classification prove the effectiveness of our proposed approach over competitive baselines.