Since the off-diagonal regions are used to store the transform information, this approach is very efficient in saving the computational memory.
If we only want to compute a few of the largest singular values and associated singular vectors of a large matrix, the Lanczos bidiagonalization is an important procedure for solving this problem [5–8].
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The singular value decomposition (SVD) is a fundamental matrix decomposition in linear algebra.
It is widely applied in many modern techniques, for example, high- dimensional data visualization, dimension reduction, data mining, latent semantic analysis, and so forth.
Although the SVD plays an essential role in these fields, its apparent weakness is the order three computational cost.
When the data configuration is Euclidean, the MDS is similar to the PCA, in that both can remove inherent noise with its compact representation of data.The singular value decomposition (SVD) and the principle component analysis (PCA) are fundamental in linear algebra and statistics.There are many modern applications based on these two tools, such as linear discriminate analysis , multidimensional scaling analysis , and feature extraction, high-dimensional data visualization.The main purpose of this paper is to deal with this problem when the numerical rank of the huge matrix is small.The second purpose of this paper is to update the SVD when the matrix size is extended by new data updating.If the matrix , in which we have proved that when the data dimension is significantly smaller than the number of data entries, there is a fast linear approach for the classical MDS.The main idea of fast MDS is using statistical resampling to split data into overlapping subsets.If the rank of matrix is much smaller than the matrix size, there are already some fast SVD approaches.In this paper, we focus on this case but with the additional condition that the data is considerably huge to be stored as a matrix form.This order three computational cost makes many modern applications infeasible, especially when the scale of the data is huge and growing.Therefore, it is imperative to develop a fast SVD method in modern era.