 # Updating the singular value decomposition

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If we compare the users based on individual movies, however, only those movies that both users have rated will affect their similarity.This is an extreme example, but one can certainly imagine that there are various classes of movies that should be compared. examined the dimensionality reduction problem in the context of information retrieval .There are several reasons we might want to do this. If we have a dataset with 17,000 movies, than each user is a vector of 17,000 coordinates, and this makes storing and comparing users relatively slow and memory-intensive.It turns out, however, that using a smaller number of dimensions can actually improve prediction accuracy.Calculating the PSVD of large term-document matrices is computationally expensive; hence in the case where terms or documents are merely added to an existing data set, it is extremely beneficial to the previously calculated PSVD to reflect the changes.It is shown how updating can be used in LSI to significantly reduce the computational cost of finding the PSVD without significantly impacting performance.Some numerical examples are given to confirm the performance of the algorithms.

wherein the step of determining updated estimates of the singular values and singular vectors includes weighting of a contribution of the orthogonal projections of the input and output vectors into the updated estimates of the singular values and singular vectors with a weighting factor of less than 1.For example, suppose we have two users who both like science fiction movies.If one user has rated highly, then it makes sense to say the users are similar.comprising a step of estimating a rank of the matrix transfer function from one of the orthogonal projections of the output vector onto the left singular vectors and the orthogonal projections of the input vector onto the right singular vectors for detecting a change of said rank.wherein the step of computing an orthogonal correction vector for each singular vector further comprises the step of computing (p−1) orthogonal expansion coefficients for each singular vector from the orthogonal projections of the output and input vectors onto initial estimates of the right and left singular vectors using the initial estimates of the singular values. 