Optimal singular value shrinkage with noise homogenization
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We derive the optimal singular values for prediction in the spiked model with noise homogenization, which equalizes the noise level across all coordinates. As part of this derivation, we obtain new asymptotic results for the high-dimensional spiked model with heteroskedastic noise, and consistent estimators for the relevant population parameters. We demonstrate the advantages of noise homogenization theoretically and through simulations. Specifically, we prove that in a certain asymptotic regime optimal shrinkage with homogenization converges to the best linear predictor, whereas shrinkage without homogenization converges to a suboptimal linear filter. We show that homogenization increases a natural signal-to-noise ratio of the observations. We also extend previous analysis on out-of-sample prediction to the setting of predictors with homogenization.
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