Support Recovery for Sparse Multidimensional Phase Retrieval

We consider the sparse phase retrieval problem of recovering a signal $\mathbf {x}$ in notation="LaTeX">$\mathbb {R}^d$ from intensity-only measurements dimension notation="LaTeX">$d \geq 2$ . Sparse can often be equivalently formulated as its autocorrelation, which is turn directly related to combinatorial set pairwise differences. In one spatial dimension, this well studied and known turnpike problem work, we present MISTR (Multidimensional Intersection supporT Recovery), an algorithm exploits formulation recover support multidimensional magnitude-only measurements. takes advantage structure multiple dimensions provably achieve same accuracy best one-dimensional algorithms dramatically less time. prove theoretically that correctly recovers signals distributed Gaussian point process with high probability long sparsity at most notation="LaTeX">$\mathcal {O}(n^{d\theta })$ for any notation="LaTeX">$\theta < 1/2$ , where notation="LaTeX">$n^d$ represents pixel size fixed image window. case magnitude are corrupted by noise, provide thresholding scheme theoretical guarantees 1/4$ obviates need explicitly handle noisy autocorrelation data. Detailed reproducible numerical experiments demonstrate effectiveness our algorithm, showing practice enjoys time complexity nearly linear input.