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.
have recently applied a leverage score-based sampling to the least-squares step of the sparse CP decomposition by showing how leverage scores of an unfolded tensor can be estimated by the leverage scores of the factor matrices @cite_17 . This approach is similar to the way that we bound the coherence of Khatri-Rao products. also use randomization within CP-ALS, specifically for the case of rank reduction, where the input to the algorithm is already in CP format @cite_43 . They use randomization to improve the conditioning of the individual least squares problems in order to compute better overall approximations. The randomization makes each iteration of their method more costly, but they observe faster convergence (and overall running time) than ALS for ill-conditioned problems.
@cite_43 @cite_17
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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. This paper introduces a randomized variation of the alternating least squares (ALS) algorithm for rank reduction of canonical tensor formats. The aim is to address the potential numerical ill-conditioning of least squares matrices at each ALS iteration. The proposed algorithm, dubbed randomized ALS, mitigates large condition numbers via projections onto random tensors, a technique inspired by well-established randomized projection methods for solving overdetermined least squares problems in a matrix setting. A probabilistic bound on the condition numbers of the randomized ALS matrices is provided, demonstrating reductions relative to their standard counterparts. Additionally, results are provided that guarantee comparable accuracy of the randomized ALS solution at each iteration. The performance of the randomized algorithm is studied with three examples, including manufactured tensors and an elliptic PDE with random inputs. In particular, for the latter, tests illustrate not only improvements in condition... We investigate an efficient parallelization of the most common iterative sparse tensor decomposition algorithms on distributed memory systems. A key operation in each iteration of these algorithms is the matricized tensor times Khatri-Rao product (MTTKRP). This operation amounts to element-wise vector multiplication and reduction depending on the sparsity of the tensor. We investigate a fine and a coarse-grain task definition for this operation, and propose hypergraph partitioning-based methods for these task definitions to achieve the load balance as well as reduce the communication requirements. We also design a distributed memory sparse tensor library, HyperTensor, which implements a well-known algorithm for the CANDECOMP- PARAFAC (CP) tensor decomposition using the task definitions and the associated partitioning methods. We use this library to test the proposed implementation of MTTKRP in CP decomposition context, and report scalability results up to 1024 MPI ranks. We observed up to 194 fold speedups using 512 MPI processes on a well-known real world data, and significantly better performance results with respect to a state of the art implementation. We present a technique for significantly speeding up Alternating Least Squares (ALS) and Gradient Descent (GD), two widely used algorithms for tensor factorization. By exploiting properties of the Khatri-Rao product, we show how to efficiently address a computationally challenging sub-step of both algorithms. Our algorithm, DFacTo, only requires two sparse matrix-vector products and is easy to parallelize. DFacTo is not only scalable but also on average 4 to 10 times faster than competing algorithms on a variety of datasets. For instance, DFacTo only takes 480 seconds on 4 machines to perform one iteration of the ALS algorithm and 1,143 seconds to perform one iteration of the GD algorithm on a 6.5 million x 2.5 million x 1.5 million dimensional tensor with 1.2 billion non-zero entries.