Matrix estimation via singular value shrinkage

In this talk, I will introduce recent studies on shrinkage estimation of matrices. First, we develop a superharmonic prior for matrices that shrinks singular values, which can be viewed as a natural generalization of Stein’s prior. This prior is motivated from the Efron–Morris estimator, which is an extension of the James–Stein estimator to matrices. The generalized Bayes estimator with respect to this prior is minimax and dominates MLE under the Frobenius loss. In particular, since it shrinks to the space of low-rank matrices, it attains large risk reduction when the unknown matrix is close to low-rank (e.g. reduced-rank regression). Next, we construct a theory of shrinkage estimation under the “matrix quadratic loss”, which is a matrix-valued loss function suitable for matrix estimation. A notion of “matrix superharmonicity” for matrix-variate functions is introduced and the generalized Bayes estimator with respect to a matrix superharmonic prior is shown to be minimax under the matrix quadratic loss. The matrix-variate improper t-priors are matrix superharmonic and this class includes the above generalization of Stein’s prior. Applications include matrix completion and nonparametric estimation.