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 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 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. The Canonical Polyadic Decomposition (CPD) of tensors is a powerful tool for analyzing multi-way data and is used extensively to analyze very large and extremely sparse datasets. The bottleneck of computing the CPD is multiplying a sparse tensor by several dense matrices. Algorithms for tensor-matrix products fall into two classes. The first class saves floating point operations by storing a compressed tensor for each dimension of the data. These methods are fast but suffer high memory costs. The second class uses a single uncompressed tensor at the cost of additional floating point operations. In this work, we bridge the gap between the two approaches and introduce the compressed sparse fiber (CSF) a data structure for sparse tensors along with a novel parallel algorithm for tensor-matrix multiplication. CSF offers similar operation reductions as existing compressed methods while using only a single tensor structure. We validate our contributions with experiments comparing against state-of-the-art methods on a diverse set of datasets. Our work uses 58 less memory than the state-of-the-art while achieving 81 of the parallel performance on 16 threads. 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.