We consider the multivariate response regression problem with a regressioncoefficient matrix of low, unknown rank. In this setting, we analyze a newcriterion for selecting the optimal reduced rank. This criterion differsnotably from the one proposed in Bunea, She and Wegkamp [7] in that it does notrequire estimation of the unknown variance of the noise, nor depends on adelicate choice of a tuning parameter. We develop an iterative, fullydata-driven procedure, that adapts to the optimal signal to noise ratio. Thisprocedure finds the true rank in a few steps with overwhelming probability. Ateach step, our estimate increases, while at the same time it does not exceedthe true rank. Our finite sample results hold for any sample size and anydimension, even when the number of responses and of covariates grow much fasterthan the number of observations. We perform an extensive simulation study thatconfirms our theoretical findings. The new method performs better and morestable than that in [7] in both low- and high-dimensional settings.
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