{"ID":2861895,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.00551","arxiv_id":"2510.00551","title":"Stable Phase Retrieval: Optimal Rates in Poisson and Heavy-tailed Models","abstract":"We investigate stable recovery guarantees for phase retrieval under two realistic and challenging noise models: the Poisson model and the heavy-tailed model. Our analysis covers both nonconvex least squares (NCVX-LS) and convex least squares (CVX-LS) estimators. For the Poisson model, we demonstrate that in the high-energy regime where the true signal $pmb{x}$ exceeds a certain energy threshold, both estimators achieve a signal-independent, minimax optimal error rate $\\mathcal{O}(\\sqrt{\\frac{n}{m}})$, with $n$ denoting the signal dimension and $m$ the number of sampling vectors. In contrast, in the low-energy regime, the NCVX-LS estimator attains an error rate of $\\mathcal{O}(\\|\\pmb{x}\\|^{1/4}_2\\cdot(\\frac{n}{m})^{1/4})$, which decreases as the energy of signal $\\pmb{x}$ diminishes and remains nearly optimal with respect to the oversampling ratio. This demonstrates a signal-energy-adaptive behavior in the Poisson setting. For the heavy-tailed model with noise having a finite $q$-th moment ($q\u003e2$), both estimators attain the minimax optimal error rate $\\mathcal{O}( \\frac{\\| ξ\\|_{L_q}}{\\| \\pmb{x} \\|_2} \\cdot \\sqrt{\\frac{n}{m}} )$ in the high-energy regime, while the NCVX-LS estimator further achieves the minimax optimal rate $\\mathcal{O}( \\sqrt{\\|ξ\\|_{L_q}}\\cdot (\\frac{n}{m})^{1/4} )$ in the low-energy regime. Our analysis builds on two key ideas: the use of multiplier inequalities to handle noise that may exhibit dependence on the sampling vectors, and a novel interpretation of Poisson noise as sub-exponential in the high-energy regime yet heavy-tailed in the low-energy regime. These insights form the foundation of a unified analytical framework, which we further apply to a range of related problems, including sparse phase retrieval, low-rank PSD matrix recovery, and random blind deconvolution.","short_abstract":"We investigate stable recovery guarantees for phase retrieval under two realistic and challenging noise models: the Poisson model and the heavy-tailed model. Our analysis covers both nonconvex least squares (NCVX-LS) and convex least squares (CVX-LS) estimators. For the Poisson model, we demonstrate that in the high-en...","url_abs":"https://arxiv.org/abs/2510.00551","url_pdf":"https://arxiv.org/pdf/2510.00551v1","authors":"[\"Gao Huang\",\"Song Li\",\"Deanna Needell\"]","published":"2025-10-01T06:11:53Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"cs.IT\"]","methods":"[]","has_code":false}