Geometry Determines Local Acceleration in Griffin-Lim Phase Retrieval

Acceleration in Griffin–Lim phase retrieval is commonly tuned heuristically, despite substantial instance-to-instance variation in local geometry. We show that this geometry determines the local acceleration problem. At a regular feasible solution, the transverse Griffin–Lim Jacobian is self-adjoint and positive semidefinite, with eigenvalues equal to the squared cosines of the principal angles between the signal-consistency subspace and the tangent space of the magnitude manifold. This geometry–spectrum correspondence turns local acceleration into an explicit spectral design problem. For the three-parameter AGLA family, we derive the exact modal dynamics of the linearization and solve the resulting minimax spectral-radius problem in closed form. The geometry-optimal design achieves a strictly smaller optimal local factor than FGLA, and hence GLA, on every nondegenerate transverse spectral interval. When the local spectrum is unknown, however, acceleration becomes an information problem. We show that passive trajectory observations cannot identify spectral directions they do not excite. Under explicit local active-subspace and consistency conditions, trajectory-based Ritz estimates nevertheless recover the excited dominant spectral edge, yielding an online method that conditionally attains the corresponding conservative geometry-designed asymptotic rate. Real-audio STFT experiments support the predicted geometry–spectrum relationship and reveal a complementary robustness–rate separation: online adaptation does not improve paired finite-window local contraction rates on average, but substantially improves empirical convergence success under the frozen initialization protocol. Across three real-audio evaluations, Online attains 73.30–74.94% success, versus 54.87–55.04% for oracle-tuned FGLA. On a disjoint 4,752-run MUSAN evaluation, the same frozen method attains 73.30%, compared with 52.06% for a literature-configured NAGLA and 47.62% for a published AGLA configuration; a larger 11,856-run MUSAN evaluation reproduces the robustness–rate separation. Together, these results identify local geometry as the organizing principle for Griffin–Lim acceleration, characterize the spectral information available to passive adaptation, and separate geometry-optimal local acceleration from what can be realized online.