The CANDECOMP PARAFAC (CP) decomposition is a leading method for the analysis of multiway data. The standard alternating least squares algorithm for the CP decomposition (CP-ALS) involves a series of highly overdetermined linear least squares problems. We extend randomized least squares methods to tensors and show the workload of CP-ALS can be drastically reduced without a sacrifice in quality. We introduce techniques for efficiently preprocessing, sampling, and computing randomized least squares on a dense tensor of arbitrary order, as well as an efficient sampling-based technique for checking the stopping condition. We also show more generally that the Khatri-Rao product (used within the CP-ALS iteration) produces conditions favorable for direct sampling. In numerical results, we see improvements in speed, reductions in memory requirements, and robustness with respect to initialization.
An alternative to sketching is to compress the tensor using lossy methods before computation. Zhou and Cichocki examine the effectiveness of performing a CP decomposition on a compressed representation of the data using the lossy Tucker decomposition to produce a smaller problem size @cite_9 . ParCube @cite_15 compresses the original tensor by directly sampling and performs a decomposition on the result.
@cite_9 @cite_15
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Tensor decompositions are promising tools for big data analytics as they bring multiple modes and aspects of data to a unified framework, which allows us to discover complex internal structures and correlations of data. Unfortunately most existing approaches are not designed to meet the major challenges posed by big data analytics. This paper attempts to improve the scalability of tensor decompositions and provides two contributions: A flexible and fast algorithm for the CP decomposition (FFCP) of tensors based on their Tucker compression; A distributed randomized Tucker decomposition approach for arbitrarily big tensors but with relatively low multilinear rank. These two algorithms can deal with huge tensors, even if they are dense. Extensive simulations provide empirical evidence of the validity and efficiency of the proposed algorithms. 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. A traffic tensor or simply origin ? destination ? time is a new data model for conventional origin destination (O D) matrices. Tensor models are traffic data analysis techniques which use this new data model to improve performance. Tensors outperform other models because both temporal and spatial fluctuations of traffic patterns are simultaneously taken into account, obtaining results that follow a more natural pattern. Three major types of fluctuations can occur in traffic tensors: mutations to the overall traffic flows, alterations to the network topology and chaotic behaviors. How can we detect events in a system that is faced with all types of fluctuations during its life cycle? Our initial studies reveal that the current design of tensor models face some difficulties in dealing with such a realistic scenario. We propose a new hybrid tensor model called HTM that enhances the detection ability of tensor models by using a parallel tracking technique on the traffic's topology. However, tensor decomposition techniques such as Tucker, a key step for tensor models, require a complicated parameter that not only is difficult to choose but also affects the model's quality. We address this problem examining a recent technique called adjustable core size Tucker decomposition (ACS-Tucker). Experiments on simulated and real-world data sets from different domains versus several techniques indicate that the proposed model is effective and robust, therefore it constitutes a viable alternative for analysis of the traffic tensors. HighlightsA new problem in tensor-based traffic models is introduced and formulated.Separation of topology makes tensor models more sensitive to topological changes.Our hybrid tensor model has 10 better detection power than naive tensor models.Tensor-based models are more robust than matrix residual methods such as DTA.Adjustable core size Tucker decomposition is a potential method for tensor models. 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.