Tucker decomposition is a low-rank tensor approximation that generalizes a truncated matrix singular value decomposition (SVD). Existing parallel software has shown that Tucker decomposition is particularly effective at compressing terabyte-sized multidimensional scientific simulation datasets, computing reduced representations that satisfy a specified approximation error. The general approach is to get a low-rank approximation of the input data by performing a sequence of matrix SVDs of tensor unfoldings, which tend to be short-fat 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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A Tensor SVD-based Classification Algorithm Applied to fMRI Data
To analyze the abundance of multidimensional data, tensor-based frameworks have been developed. Traditionally, the matrix singular value decomposition (SVD) is used to extract the most dominant features from a matrix containing the vectorized data. While the SVD is highly useful for data that can be appropriately represented as a matrix, this step of vectorization causes us to lose the high-dimensional relationships intrinsic to the data. To facilitate efficient multidimensional feature extraction, we utilize a projection-based classification algorithm using the t-SVDM, a tensor analog of the matrix SVD. Our work extends the t-SVDM framework and the classification algorithm, both initially proposed for tensors of order 3, to any number of dimensions. We then apply this algorithm to a classification task using the StarPlus fMRI dataset. Our numerical experiments demonstrate that there exists a superior tensor-based approach to fMRI classification than the best possible equivalent matrix-based approach. Our results illustrate the advantages of our chosen tensor framework, provide insight into beneficial choices of parameters, and could be further developed for classification of more complex imaging data. We provide our Python implementation at https://github.com/elizabethnewman/tensor-fmri
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- NSF-PAR ID:
- 10326606
- Date Published:
- Journal Name:
- ArXivorg
- ISSN:
- 2331-8422
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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