|
Abstract:
|
In this paper, we examine the effects of the group information and the proportion of $\ell_1$ penalty $(\gamma)$ on various performance measures, for both the noisy and noiseless cases, under any sampling rate ($n/p$).We demonstrate that there is a trade-off between the type I and II errors along the SGL path. More specifically, we derive the sharp asymptotic trade-off between true positive proportion (TPP) and false discovery proportion (FDP), consisting of possibly multiple line segments. Similar to LASSO, SGL also suffers from the Donoho-Tanner phase transition (a TPP upper bound) but may additionally have a TPP lower bound. With sufficiently correct group information and smaller $\gamma$, we show that SGL can have significantly better performance measure than LASSO, in terms of TPP, FDP and mean squared error. Furthermore, we investigate the effect of $\gamma$ when the group information is imperfect. Our theoretical analysis leverages the approximate message passing (AMP) theory and properties of the SGL proximal operator. Finally, simulations and real-data experiments are presented to support our results.
|
