Recent years have witnessed the world-wide emergence of mega-metropolises with incredibly huge populations. Understanding residents mobility patterns, or urban dynamics, thus becomes crucial for building modern smart cities. In this paper, we propose a Neighbor-Regularized and context-aware Non-negative Tensor Factorization model (NR-cNTF) to discover interpretable urban dynamics from urban heterogeneous data. Different from many existing studies concerned with prediction tasks via tensor completion, NR-cNTF focuses on gaining urban managerial insights from spatial, temporal, and spatio-temporal patterns. This is enabled by high-quality Tucker factorizations regularized by both POI-based urban contexts and geographically neighboring relations. NR-cNTF is also capable of unveiling long-term evolutions of urban dynamics via a pipeline initialization approach. We apply NR-cNTF to a real-life data set containing rich taxi GPS trajectories and POI records of Beijing. The results indicate: 1) NR-cNTF accurately captures four kinds of city rhythms and seventeen spatial communities; 2) the rapid development of Beijing, epitomized by the CBD area, indeed intensifies the job-housing imbalance; 3) the southern areas with recent government investments have shown more healthy development tendency. Finally, NR-cNTF is compared with some baselines on traffic prediction, which further justifies the importance of urban contexts awareness and neighboring regulations.
Among the abundant methods for human mobility data mining, tensor factorization decomposition, like CANDECOMP PARAFAC (CP) @cite_9 and Tucker factorizations @cite_2 , gains particular interests for its distinct ability in modeling multi-aspect heterogeneous big data. Indeed, in city scenarios data samples are always involved with many aspects, such as time, space, human, urban contexts and so on, and therefore are very suitable for tensor factorization based data mining methods @cite_11 . Typical applications of tensor factorization could be classified into two categories. The first category is to reconstruct tensors for predicting unknown values in multi-aspect data sets, such as completing missing traffic data @cite_23 , inferring urban gas consumption @cite_31 , predicting travel time @cite_25 , recommending social tags @cite_21 , movies @cite_30 and sightseeing locations @cite_35 @cite_34 , and so on.
@cite_30 @cite_35 @cite_9 @cite_21 @cite_23 @cite_2 @cite_31 @cite_34 @cite_25 @cite_11
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CANDECOMP PARAFAC (CP) tensor factorization of incomplete data is a powerful technique for tensor completion through explicitly capturing the multilinear latent factors. The existing CP algorithms require the tensor rank to be manually specified, however, the determination of tensor rank remains a challenging problem especially for CP rank . In addition, existing approaches do not take into account uncertainty information of latent factors, as well as missing entries. To address these issues, we formulate CP factorization using a hierarchical probabilistic model and employ a fully Bayesian treatment by incorporating a sparsity-inducing prior over multiple latent factors and the appropriate hyperpriors over all hyperparameters, resulting in automatic rank determination. To learn the model, we develop an efficient deterministic Bayesian inference algorithm, which scales linearly with data size. Our method is characterized as a tuning parameter-free approach, which can effectively infer underlying multilinear factors with a low-rank constraint, while also providing predictive distributions over missing entries. Extensive simulations on synthetic data illustrate the intrinsic capability of our method to recover the ground-truth of CP rank and prevent the overfitting problem, even when a large amount of entries are missing. Moreover, the results from real-world applications, including image inpainting and facial image synthesis, demonstrate that our method outperforms state-of-the-art approaches for both tensor factorization and tensor completion in terms of predictive performance. Tensor CANDECOMP PARAFAC (CP) decomposition has wide applications in statistical learning of latent variable models and in data mining. In this paper, we propose fast and randomized tensor CP decomposition algorithms based on sketching. We build on the idea of count sketches, but introduce many novel ideas which are unique to tensors. We develop novel methods for randomized computation of tensor contractions via FFTs, without explicitly forming the tensors. Such tensor contractions are encountered in decomposition methods such as tensor power iterations and alternating least squares. We also design novel colliding hashes for symmetric tensors to further save time in computing the sketches. We then combine these sketching ideas with existing whitening and tensor power iterative techniques to obtain the fastest algorithm on both sparse and dense tensors. The quality of approximation under our method does not depend on properties such as sparsity, uniformity of elements, etc. We apply the method for topic modeling and obtain competitive results. Tensor (multiway array) factorization and decomposition has become an important tool for data mining. Fueled by the computational power of modern computer researchers can now analyze large-scale tensorial structured data that only a few years ago would have been impossible. Tensor factorizations have several advantages over two-way matrix factorizations including uniqueness of the optimal solution and component identification even when most of the data is missing. Furthermore, multiway decomposition techniques explicitly exploit the multiway structure that is lost when collapsing some of the modes of the tensor in order to analyze the data by regular matrix factorization approaches. Multiway decomposition is being applied to new fields every year and there is no doubt that the future will bring many exciting new applications. The aim of this overview is to introduce the basic concepts of tensor decompositions and demonstrate some of the many benefits and challenges of modeling data multiway for a wide variety of data and problem domains. © 2011 John Wiley & Sons, Inc. WIREs Data Mining Knowl Discov 2011 1 24-40 DOI: 10.1002 widm.1 Tensor CANDECOMP PARAFAC (CP) decomposition is a powerful but computationally challenging tool in modern data analytics. In this paper, we show ways of sampling intermediate steps of alternating minimization algorithms for computing low rank tensor CP decompositions, leading to the sparse alternating least squares (SPALS) method. Specifically, we sample the the Khatri-Rao product, which arises as an intermediate object during the iterations of alternating least squares. This product captures the interactions between different tensor modes, and form the main computational bottleneck for solving many tensor related tasks. By exploiting the spectral structures of the matrix Khatri-Rao product, we provide efficient access to its statistical leverage scores. When applied to the tensor CP decomposition, our method leads to the first algorithm that runs in sublinear time per-iteration and approximates the output of deterministic alternating least squares algorithms. Empirical evaluations of this approach show significantly speedups over existing randomized and deterministic routines for performing CP decomposition. On a tensor of the size 2.4m by 6.6m by 92k with over 2 billion nonzeros formed by Amazon product reviews, our routine converges in two minutes to the same error as deterministic ALS.