Randomized CP Tensor Decomposition

N. Benjamin Erichson, Krithika Manohar, Steven L. Brunton, J. Nathan Kutz

The CANDECOMP/PARAFAC (CP) tensor decomposition is a popular dimensionality-reduction method for multiway data. Dimensionality reduction is often sought since many high-dimensional tensors have low intrinsic rank relative to the dimension of the ambient measurement space. However, the emergence of `big data' poses significant computational challenges for computing this fundamental tensor decomposition. Leveraging modern randomized algorithms, we demonstrate that the coherent structure can be learned from a smaller representation of the tensor in a fraction of the time. Moreover, the high-dimensional signal can be faithfully approximated from the compressed measurements. Thus, this simple but powerful algorithm enables one to compute the approximate CP decomposition even for massive tensors. The approximation error can thereby be controlled via oversampling and the computation of power iterations. In addition to theoretical results, several empirical results demonstrate the performance of the proposed algorithm.

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