We analyze its properties and show its potential for exploring graph structures. We prove that under proper absorption rates, a random walk starting from a set S of low conductance will be mostly absorbed in S. Moreover, the absorption probabilities vary slowly inside S, while dropping sharply outside, thus implementing the desirable cluster assumption for graph-based learning. Remarkably, the partially absorbing process unifies many popular models arising in a variety of contexts, provides new insights into them, and makes it possible for transferring findings from one paradigm to another. Simulation results demonstrate its promising applications in retrieval and classification.

We find that various well-known graph-based models exhibit a common important harmonic structure in its target function - the value of a vertex is approximately the weighted average of the values of its adjacent neighbors. Understanding of such structure and analysis of the loss defined over such structure help reveal important properties of the target function over a graph. In this paper, we show that the variation of the target function across a cut can be upper and lower bounded by the ratio of its harmonic loss and the cut cost. We use this to develop an analytical tool and analyze five popular graph-based models: absorbing random walks, partially absorbing random walks, hitting times, pseudo-inverse of the graph Laplacian, and eigenvectors of the Laplacian matrices. Our analysis sheds new insights into several open questions related to these models, and provides theoretical justifications and guidelines for their practical use. Simulations on synthetic and real datasets confirm the potential of the proposed theory and tool.

Modern vehicles are equipped with increasingly complex sensors. These sensors generate large volumes of data that provide opportunities for modeling and analysis. Here, we are interested in exploiting this data to learn aspects of behaviors and the road network associated with individual drivers. Our dataset is collected on a standard vehicle used to commute to work and for personal trips. A Hidden Markov Model (HMM) trained on the GPS position and orientation data is utilized to compress the large amount of position information into a small amount of road segment states. Each state has a set of observations, i.e. car signals, associated with it that are quantized and modeled as draws from a Hierarchical Dirichlet Process (HDP). The inference for the topic distributions is carried out using HDP split-merge sampling algorithm. The topic distributions over joint quantized car signals characterize the driving situation in the respective road state. In a novel manner, we demonstrate how the sparsity of the personal road network of a driver in conjunction with a hierarchical topic model allows data driven predictions about destinations as well as likely road conditions.

We show that either explicitly or implicitly, various well-known graph-based models exhibit a common significant \emph{harmonic} structure in its target function -- the value of a vertex is approximately the weighted average of the values of its adjacent neighbors. Understanding of such structure and analysis of the loss defined over such structure help reveal important properties of the target function over a graph. In this paper, we show that the variation of the target function across a cut can be upper and lower bounded by the ratio of its harmonic loss and the cut cost. We use this to develop an analytical tool and analyze 5 popular models in graph-based learning: absorbing random walks, partially absorbing random walks, hitting times, pseudo-inverse of graph Laplacian, and eigenvectors of the Laplacian matrices. Our analysis well explains several open questions of these models reported in the literature. Furthermore, it provides theoretical justifications and guidelines for their practical use. Simulations on synthetic and real datasets support our analysis.

We propose a novel stochastic process that is with probability $\alpha_i$ being absorbed at current state $i$, and with probability $1-\alpha_i$ follows a random edge out of it. We analyze its properties and show its potential for exploring graph structures. We prove that under proper absorption rates, a random walk starting from a set $\mathcal{S}$ of low conductance will be mostly absorbed in $\mathcal{S}$. Moreover, the absorption probabilities vary slowly inside $\mathcal{S}$, while dropping sharply outside $\mathcal{S}$, thus implementing the desirable cluster assumption for graph-based learning. Remarkably, the partially absorbing process unifies many popular models arising in a variety of contexts, provides new insights into them, and makes it possible for transferring findings from one paradigm to another. Simulation results demonstrate its promising applications in graph-based learning.