Stable multi-wavelet phase retrieval under discrete sampling

We study the problem of wavelet phase retrieval, that is, the reconstruction of a signal f ∈ L 2 ( R ) from phaseless samples of its wavelet transform taken on a hyperbolic lattice. While uniqueness is known for signals under restrictive structural assumptions, the general case of complex-valued signals remains open, and stability under discrete sampling is largely unexplored. Motivated by recent advances in multi-measurement phase retrieval (Grohs et al., 2025; Alaifari et al., 2024), we propose a multi-wavelet sampling framework. Our approach uses phaseless measurements from eight specially designed wavelets ( ϕ j ) j = 1 8 , sampled on a hyperbolic lattice. We prove that every signal f ∈ L 2 ( R ) satisfying a mild connectivity condition is uniquely determined, up to a global phase factor, by { | W ϕ j f ( α 0 n , β 0 m α 0 n ) | : n , m ∈ Z , j = 1 , … , 8 } with suitable sampling parameters α 0 , β 0 . Beyond uniqueness, we adopt the relaxed notion of stability proposed for phase retrieval in infinite-dimensional spaces (Alaifari et al., 2019), and establish corresponding stability results for signals whose wavelet transform coefficients are concentrated in one or more localized regions of the sampling lattice. Taken together, our results provide the first discrete sampling scheme that guarantees both uniqueness and stability for a broad class of complex-valued signals in wavelet phase retrieval, substantially advancing its theoretical foundations toward practical applicability.