Fast Proper Orthogonal Decomposition Using Improved Sampling and Iterative Techniques for Singular Value Decomposition
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Proper Orthogonal Decomposition (POD), also known as Principal component analysis (PCA), is a dimensionality reduction technique used to capture the energetically dominant features of datasets, known as eigenfeatures or POD modes. These modes can be obtained by finding a low rank approximation of the data matrix using singular value decomposition (SVD). In this paper, we explore random sampling techniques, which is one approach to obtain an approximate low rank description of the dataset in a computationally efficient manner. We analyse the performance of two algorithms proposed by [1], namely LTSVD and CTSVD, and discuss their advantages and limitations. We modify the two algorithms to improve the runtime of the methods and prove the equivalence of our modified algorithms with LTSVD and CTSVD. The modifications we propose are independent of sampling probability distributions and can be used to improve runtime whenever sampling is done with replacement. We use the modified algorithms along with a previously proposed merging technique to obtain the SVD of large matrices that do not fit in memory. We also propose an iterative algorithm to improve the approximation of the POD modes and the subspace spanned by the modes. Unlike the previous methods for multiple rounds of sampling, we obtain an updated approximation to the POD modes in each iteration and stop when the modes or subspace spanned by the modes have converged. The performance of our proposed solutions is analysed using four datasets of various sizes for single and multi-threaded execution. In all cases, we obtain a significant speedup over using a truncated SVD. The speedup of our modified LTSVD and CTSVD algorithms with respect to the existing algorithms depends on the error parameters. For low values of error parameters, we get upto 2-3x speedup. The results of the iterative algorithms have significantly better accuracies.
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