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.
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].
One of the key approaches of the MDS is simply the SVD, that is, if we can find a fast approach of the MDS then it is possible to find a fast approach of the SVD.
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.
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In this paper, we would like to implement SCMDS to the fast SVD approach, say SCSVD.The following subsections are reviews of the classical MDS and the SCSVD.Assume that computational complexity makes it infeasible.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.In recent years, digital information has been proliferating and many analytic methods based on the PCA and the SVD are facing the challenge of their significant computational cost.Thus, it is crucial to develop a fast approach to compute the PCA and the SVD.Currently there are some well-known methods for computing the SVD.For example, the GR-SVD is a two-step method which performs Householder transformations to reduce the matrix to bidiagonal form then performs the QR iteration to obtain the singular values [3, 4].We will demonstrate that this fast SVD result is sufficiently accurate, and most importantly it can be derived immediately.Using this fast method, many infeasible modern techniques based on the SVD will become viable.