Diagonal-Preserving Power-Coherence Parameterization for Learned Covariance Source Imaging

Learned estimators for coherent acoustic imaging predict a source power map from microphone cross-spectral data, but the measurements depend on the full source covariance, of which the map is only the diagonal. We study how a learned estimate should be refined against the measurements within a fixed optimization budget, and show that the choice of covariance coordinates matters. The conventional diagonal-plus-low-rank form couples power and coherence, so updating its coherent factor also changes the map. We adapt a bounded factor-correlation model to complex covariance: the map remains the covariance diagonal, and positive semidefiniteness holds without a dense projection. Both forms describe the same trace-normalized covariance family and can start from the same physical covariance, so matched controls isolate the effect of coordinates from learned initialization and learning rate selection. With separate development-selected rates for power and factors, power–coherence (PC) refinement lowers shape error by 26.5–30.0% and raises spatial overlap by 11.3–14.6 percentage points after 80 updates without a map anchor, across eight paired seeds and three initializations. The largest gains from CNN starts occur on coherent lines and arcs, with consistent gains on synthetic blade mixtures. With an anchor, PC lowers the joint objective by 17.4–18.6%. An independent time-domain simulator also shows spatial gains without retuning, while Gaussian and Fourier operators show objective reductions. These results identify optimization coordinates as a design choice for refining learned estimates in covariance inverse problems, even when the conventional form attains a lower measurement residual.