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The Gaussian sequence model is a canonical model in nonparametric estimation.In this study, we introduce a multivariate version of the Gaussian sequencemodel and investigate adaptive estimation over the multivariate Sobolevellipsoids, where adaptation is not only to unknown smoothness but also toarbitrary quadratic loss. First, we derive an oracle inequality for thesingular value shrinkage estimator by Efron and Morris, which is a matrixgeneralization of the James--Stein estimator. Next, we develop anasymptotically minimax estimator on the multivariate Sobolev ellipsoid for eachquadratic loss, which can be viewed as a generalization of Pinsker's theorem.Then, we show that the blockwise Efron--Morris estimator is exactly adaptiveminimax over the multivariate Sobolev ellipsoids under the correspondingquadratic loss. It attains sharp adaptive estimation of any linear combinationof the mean sequences simultaneously.

Adapting to general quadratic loss via singular value shrinkage