Theoretical exact-recovery guarantees for multiscale GSPWD

Establish theoretical exact-recovery guarantees for the multiscale generalized sparse plane wave decomposition approach and its sparse-recovery procedure.

Background

The paper introduces generalized sparse plane wave decomposition (GSPWD), which replaces spherical Dirichlet kernels with optimized spherical kernels that suppress sidelobe and backlobe power, and then extends the method to a multiscale formulation that jointly processes generalized steered response functions across spherical-harmonic orders. Numerical simulations indicate improved support recovery relative to single-scale and baseline SPWD methods. However, the paper does not derive formal conditions under which the proposed multiscale method exactly recovers the plane-wave support or amplitudes. Such guarantees would provide theoretical validation beyond the empirical results and could characterize the roles of dictionary coherence, sparsity, spherical-harmonic order, grid resolution, and noise.

The unresolved issue is specifically the absence of a theoretical exact-recovery analysis for the proposed approach, rather than a request to improve its empirical performance.