Event detection is gaining increasing attention in smart cities research. Large-scale mobility data serves as an important tool to uncover the dynamics of urban transportation systems, and more often than not the dataset is incomplete. In this article, we develop a method to detect extreme events in large traffic datasets, and to impute missing data during regular conditions. Specifically, we propose a robust tensor recovery problem to recover low rank tensors under fiber-sparse corruptions with partial observations, and use it to identify events, and impute missing data under typical conditions. Our approach is scalable to large urban areas, taking full advantage of the spatio-temporal correlations in traffic patterns. We develop an efficient algorithm to solve the tensor recovery problem based on the alternating direction method of multipliers (ADMM) framework. Compared with existing @math norm regularized tensor decomposition methods, our algorithm can exactly recover the values of uncorrupted fibers of a low rank tensor and find the positions of corrupted fibers under mild conditions. Numerical experiments illustrate that our algorithm can exactly detect outliers even with missing data rates as high as 40 , conditioned on the outlier corruption rate and the Tucker rank of the low rank tensor. Finally, we apply our method on a real traffic dataset corresponding to downtown Nashville, TN, USA and successfully detect the events like severe car crashes, construction lane closures, and other large events that cause significant traffic disruptions.
Low rank matrix and tensor learning has been widely used to utilize the inner structure of the data. Various application have benefited from matrix and tensor based methods, including data completion @cite_43 @cite_47 , link prediction @cite_33 , network structure clustering @cite_8 , etc.
@cite_43 @cite_47 @cite_33 @cite_8
2014237985 2744847173 2084983808 2951021721
The low-rank matrix completion problem is a fundamental machine learning problem with many important applications. The standard low-rank matrix completion methods relax the rank minimization problem by the trace norm minimization. However, this relaxation may make the solution seriously deviate from the original solution. Meanwhile, most completion methods minimize the squared prediction errors on the observed entries, which is sensitive to outliers. In this paper, we propose a new robust matrix completion method to address these two problems. The joint Schatten @math -norm and @math -norm are used to better approximate the rank minimization problem and enhance the robustness to outliers. The extensive experiments are performed on both synthetic data and real world applications in collaborative filtering and social network link prediction. All empirical results show our new method outperforms the standard matrix completion methods. Low rank tensor representation underpins much of recent progress in tensor completion. In real applications, however, this approach is confronted with two challenging problems, namely (1) tensor rank determination; (2) handling real tensor data which only approximately fulfils the low-rank requirement. To address these two issues, we develop a data-adaptive tensor completion model which explicitly represents both the low-rank and non-low-rank structures in a latent tensor. Representing the non-low-rank structure separately from the low-rank one allows priors which capture the important distinctions between the two, thus enabling more accurate modelling, and ultimately, completion. Through defining a new tensor rank, we develop a sparsity induced prior for the low-rank structure, with which the tensor rank can be automatically determined. The prior for the non-low-rank structure is established based on a mixture of Gaussians which is shown to be flexible enough, and powerful enough, to inform the completion process for a variety of real tensor data. With these two priors, we develop a Bayesian minimum mean squared error estimate (MMSE) framework for inference which provides the posterior mean of missing entries as well as their uncertainty. Compared with the state-of-the-art methods in various applications, the proposed model produces more accurate completion results. The low-rank matrix completion problem is a fundamental machine learning and data mining problem with many important applications. The standard low-rank matrix completion methods relax the rank minimization problem by the trace norm minimization. However, this relaxation may make the solution seriously deviate from the original solution. Meanwhile, most completion methods minimize the squared prediction errors on the observed entries, which is sensitive to outliers. In this paper, we propose a new robust matrix completion method to address these two problems. The joint Schatten @math p -norm and @math l p -norm are used to better approximate the rank minimization problem and enhance the robustness to outliers. The extensive experiments are performed on both synthetic data and real-world applications in collaborative filtering prediction and social network link recovery. All empirical results show that our new method outperforms the standard matrix completion methods. Recovering a low-rank tensor from incomplete information is a recurring problem in signal processing and machine learning. The most popular convex relaxation of this problem minimizes the sum of the nuclear norms of the unfoldings of the tensor. We show that this approach can be substantially suboptimal: reliably recovering a @math -way tensor of length @math and Tucker rank @math from Gaussian measurements requires @math observations. In contrast, a certain (intractable) nonconvex formulation needs only @math observations. We introduce a very simple, new convex relaxation, which partially bridges this gap. Our new formulation succeeds with @math observations. While these results pertain to Gaussian measurements, simulations strongly suggest that the new norm also outperforms the sum of nuclear norms for tensor completion from a random subset of entries. Our lower bound for the sum-of-nuclear-norms model follows from a new result on recovering signals with multiple sparse structures (e.g. sparse, low rank), which perhaps surprisingly demonstrates the significant suboptimality of the commonly used recovery approach via minimizing the sum of individual sparsity inducing norms (e.g. @math , nuclear norm). Our new formulation for low-rank tensor recovery however opens the possibility in reducing the sample complexity by exploiting several structures jointly.