Deflection-Optimal Spectral Design for Diagonal Screening in Sparse Phase Retrieval Initialization

Spectral initialization is a critical yet challenging step in sparse phase retrieval. Existing spectral design theory is largely tailored to dense phase retrieval, where the objective is eigenvector estimation. In contrast, sparse initialization first requires a statistically distinct support screening step whose design remains much less understood. This paper develops a stage-specific design theory for diagonal support screening. We formulate each coordinate score as a scalar statistic for distinguishing support from non-support coordinates and adopt the deflection criterion as a tractable measure of screening quality. Within a Hilbert-space formulation, we characterize the optimal spectral preprocessors that maximize this criterion. In the Gaussian model, the unique optimum is the centered linear preprocessor. To obtain a bounded implementation, we introduce a spherical normalization and characterize its exact optimal preprocessor. Since the exact spherical optimum exhibits a boundary singularity, we construct a bounded surrogate preprocessor and establish its unique optimality under a surrogate deflection criterion. The surrogate optimum is shown to be the direction-only projection of the Gaussian rule, removing the unbounded radial factor while preserving the same first-order screening structure. We further establish a general finite-sample diagonal bridge that connects the exact and surrogate deflection quotients to the initialization sample complexity, and that replacing the unknown signal energy by its empirical estimate introduces only a lower-order perturbation. Numerical experiments are consistent with the ordering predicted by the design quotients and show that the Gaussian centered rule and its spherical counterpart behave nearly identically at both the screening and initialization levels.