Shrinkage to smooth non-convex cone :Principal component analysis as stein estimation

In Kuriki and Takemura [1997a) we established a general theory of James-Stein type shrinkage to convex sets with smooth boundary. In this paper we show that our results can be generalized to the case where shrinkage is toward smooth non-convex cones. A primary example of this shrinkage is descriptive principal component analysis, where one shrinks small singular values of the data matrix. Here principal component analysis is interpreted as the problem of estimation of matrix mean and the shrinkage of the small singular values is regarded as shrinkage of the data matrix toward the manifold of matrices of smaller rank.