# Updating the singular value decomposition

Since incomplete data does not uniquely specify an SVD, the procedure selects one having minimal rank.For a dense p q matrix of low rank r, the incremental method has time complexity O(pqr) and space complexity O((p q)r)---better than highly optimized batch algorithms such as MATLAB 's svd().The SVD is developed as data arrives, and can handle arbitrary missing/untrusted values, correlated uncertainty across rows or columns of the measurement matrix, and user priors.Since incomplete data does not uniq ..." We introduce an incremental singular value decomposition (SVD) of incomplete data.Previous algorithms can be unstable and compute the singular values and the singular vectors of A 0 in O \Gamma (m n) min 2 (m;n) \Delta floating point operations. The columns of U and V are the left singular vectors and the right singular vectors of A, respectively; the diagonal entries of\Omega are the singular values of A... We introduce an incremental singular value decomposition (SVD) of incomplete data.In cases of missing data, it produces factorings of lower rank and residual than batch SVD algorithms applied to standard missing-data imputations.We show applications in computer vision and audio feature extraction.

We present stable and fast algorithms for computing the singular values and the singular vectors of A 0 in O \Gamma (m n) min(m;n) log 2 2 ffl \Delta floating point operations, where ffl is the machine precision. The singular value decomposition (SVD) of a matrix A 2 R m\Thetan is A = U\Omega V T ; (1.1) where U 2 R m\Thetam and V 2 R n\Thetan are orthonormal; and\Omega 2 R m\Thetan is zero except on the main diagonal, which has non-negative entries in decreasing order.SPIE 2296, Advanced Signal Processing: Algorithms, Architectures, and Implementations V, (28 October 1994); https://doi.org/10.1117/12.190853 Flavio Lorenzelli, Kung Yao, "Updating rate of Jacobi singular value decomposition (SVD) arrays and data nonstationarity," Proc.For this package to work only Numpy, Scipy and Matplotlib are required. However, Scipy need to be compiled from sources in order to use some LAPACK function "dlasd4" which are not exposed originally.In the `variable rotational rate' scheme, the number of Jacobi rotations per update is dynamically determined.The alternative approach is to make the forgetting factor variable and data-dependent.

We present stable and fast algorithms for computing the singular values and the singular vectors of A 0 in O \Gamma (m n) min(m;n) log 2 2 ffl \Delta floating point operations, where ffl is the machine precision. The singular value decomposition (SVD) of a matrix A 2 R m\Thetan is A = U\Omega V T ; (1.1) where U 2 R m\Thetam and V 2 R n\Thetan are orthonormal; and\Omega 2 R m\Thetan is zero except on the main diagonal, which has non-negative entries in decreasing order.

SPIE 2296, Advanced Signal Processing: Algorithms, Architectures, and Implementations V, (28 October 1994); https://doi.org/10.1117/12.190853 Flavio Lorenzelli, Kung Yao, "Updating rate of Jacobi singular value decomposition (SVD) arrays and data nonstationarity," Proc.

For this package to work only Numpy, Scipy and Matplotlib are required. However, Scipy need to be compiled from sources in order to use some LAPACK function "dlasd4" which are not exposed originally.

In the `variable rotational rate' scheme, the number of Jacobi rotations per update is dynamically determined.

The alternative approach is to make the forgetting factor variable and data-dependent.

In this paper, we are concerned with the behavior of this algorithm for nonstationary data, and the effect of the pipeline rate on tracking accuracy.