Dimensionality Reduction - Stanford University

Chapter 11 Dimensionality ReductionThere are many sources of data that can be viewed as a large matrix. Wesaw in Chapter 5 how the Web can be represented as a transition matrix. InChapter 9, the utility matrix was a point of focus. And in Chapter 10 weexamined matrices that represent social networks. In many of these matrixapplications, the matrix can be summarized by finding narrower matricesthat in some sense are close to the original. These narrow matrices have only asmall number of rows or a small number of columns, and therefore can be usedmuch more efficiently than can the original large matrix. The processof findingthese narrow matrices is calleddimensionality saw a preliminary example of Dimensionality Reduction in Section , we discussed UV-decomposition of a matrix and gave a simple algorithmfor finding this decomposition. Recall that a large matrixMwas decomposedinto two matricesUandVwhose productU Vwas approximatelyM. ThematrixUhad a small number of columns whereasVhad a small number of rows,so each was significantly smaller thanM, and yet together they representedmost of the information inMthat was useful in predicting ratings of items this chapter we shall explore the idea of Dimensionality Reduction inmore detail.

nonzero vector x0 and then iterate: xk+1:= Mxk kMxkk where kNk for a matrix or vector N denotes the Frobenius norm; that is, the square root of the sum of the squares of the elements of N. We multiply the current vector xk by the matrix M until convergence (i.e., kxk − xk+1k is less than some small, chosen constant). Let x be xk for that ...

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