how to generate random isotropic vectors using R?

How can I approach selecting effective vectors using Singular Value Decomposition?

• I have data matrix X where each column is an individual observation. I ran SVD and generated U,S and V matrices such that .[math] X=USV^T [/math]. Now I want to select some columns from X that has eigen vectors closer to eigen vectors of X. How can I approach to this problem ?

• Answer:

  You seem to be describing either sparse principal components (see Bair & Tibshirani 2005 for the supervised version, or Witten & Tibshirani 2008 & 2009 for unsupervised versions, for a few of many references to this topic) or the selection of K non-noise singular vectors via (e.g.) parsimony or random matrix theory (see e.g. Teschendorff 2010 for a reference to the latter). The former is relatively easier to justify than the latter, but choosing the appropriate dimensionality for a low-rank representation of a high-dimensional data matrix is not a settled matter. There is an elegant and intuitive appeal to the probabilistic interpretation of PCA as described in Tipping & Bishop (http://www.robots.ox.ac.uk/~cvrg/hilary2006/ppca.pdf ) especially when coupled with a variational approximation to the posterior distribution.  But that doesn't always produce substantially better results than the heuristics described. (For tiny data matrices you can brute force the matter iteratively, but for tiny matrices this is not a major issue :-))