The CP tensor decomposition is used in applications such as machine learning and signal processing to discover latent lowrank structure in multidimensional data. Computing a CP decomposition via an alternating least squares (ALS) method reduces the problem to several linear least squares problems. The standard way to solve these linear least squares subproblems is to use the normal equations, which inherit special tensor structure that can be exploited for computational efficiency. However, the normal equations are sensitive to numerical illconditioning, which can compromise the results of the decomposition. In this paper, we develop versions of the CPALS algorithm using the QR decomposition and the singular value decomposition, which are more numerically stable than the normal equations, to solve the linear least squares problems. Our algorithms utilize the tensor structure of the CPALS subproblems efficiently, have the same complexity as the standard CPALS algorithm when the input is dense and the rank is small, and are shown via examples to produce more stable results when illconditioning is present. Our MATLAB implementation achieves the same running time as the standard algorithm for small ranks, and we show that the new methods can obtain lower approximation error.
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Parallel Tucker Decomposition with Numerically Accurate SVD
Tucker decomposition is a lowrank tensor approximation that generalizes a truncated matrix singular value decomposition (SVD). Existing parallel software has shown that Tucker decomposition is particularly effective at compressing terabytesized multidimensional scientific simulation datasets, computing reduced representations that satisfy a specified approximation error. The general approach is to get a lowrank approximation of the input data by performing a sequence of matrix SVDs of tensor unfoldings, which tend to be shortfat matrices. In the existing approach, the SVD is performed by computing the eigendecomposition of the Gram matrix of the unfolding. This method sacrifices some numerical stability in exchange for lower computation costs and easier parallelization. We propose using a more numerically stable though more computationally expensive way to compute the SVD by pre processing with a QR decomposition step and computing an SVD of only the small triangular factor. The more numerically stable approach allows us to achieve the same accuracy with half the working precision (for example, single rather than double precision). We demonstrate that our method scales as well as the existing approach, and the use of lower precision leads to an overall reduction in running time of up to a factor of 2 when using 10s to 1000s of processors. Using the same working precision, we are also able to compute Tucker decompositions with much smaller approximation error.
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 Award ID(s):
 1942892
 NSFPAR ID:
 10318139
 Date Published:
 Journal Name:
 50th International Conference on Parallel Processing
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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