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We study a class of real robust phase retrieval problems under a Gaussian assumption on the coding matrix when the received signal is sparsely corrupted by noise. The goal is to establish conditions on the sparsity under which the input vector can be exactly recovered. The recovery problem is formulated as residual minimization in the $\ell _{1}$ -norm. The main contribution is a robust phase retrieval counterpart to the seminal paper by Candes and Tao on compressed sensing ($\ell _{1}$ regression) [Decoding by linear programming. IEEE Transactions on Information Theory, 51(12):4203–4215, 2005]. The analysis depends on a key new property of the coding matrix called the Absolute Range Property (ARP) which is the analogue to the Null Space Property (NSP) in compressed sensing. When the residuals are computed using squared magnitudes, we show that ARP follows from a standard Restricted Isometry Property (RIP). However, when the residuals are computed using absolute magnitudes, a different kind of RIP or growth property is required. We conclude by showing that the robust phase retrieval objectives are sharp with respect to their minimizers with high probability. [ABSTRACT FROM AUTHOR]

Rising CO 2 concentration in the atmosphere may modulate the response of plants to water and nitrogen, and the existing research is not sufficient to support the fine management of water and nitrogen under CO 2 rise. To determine the optimal water and nitrogen strategy for melon production under CO 2 enrichment in the future, we conducted an experiment with 12 treatments over three growing seasons. The effects of water nitrogen coupling on the yield, quality, and water−nitrogen use efficiency of oriental melon (Cucumis melo L. Qianyu No. 6) under CO 2 enrichment were investigated and evaluated comprehensively. The treatments included two CO 2 concentrations, C 1 (400 ppm) and C 2 (800 ppm), three irrigation levels, I 1 (80% evaporation (Ep)), I 2 (100% Ep), and I 3 (120% Ep); two nitrogen levels, N 1 (S: 184.77 kg ha–1, F: 147.21 kg ha–1) and N 2 (S: 369.54 kg ha–1, F: 294.42 kg ha–1). The result showed that the yield increased with increasing levels of irrigation and nitrogen, and enriched CO 2 significantly increased the yield by 10.3 − 22.9%. N 2 showed a significant promoting effect on the ratio of marketable fruit in the two spring seasons. Enriched CO 2 significantly increased free amino acid of I 1. C 2 coupled with I 1 significantly reduced nitrate content by 30.9%, 14.9% in 2020 F and 2021 S, respectively. The interaction of C*I was significant in irrigation water−use efficiency (IWUE) and nitrogen−use efficiency (NUE). In particular, C 2 coupled with I 2 increased IWUE by 40.5%, 18.7% and NUE by 43.9%, 13.9% in the two spring seasons, respectively. Ten indicators from three categories of yield, quality, and efficiency were used to establish the evaluation system of melon and were comprehensively evaluated based on Fuzzy−Borda model combined three single models. Based on the combined evaluation model, intermediate irrigation combined with higher nitrogen input was best for melon production under CO 2 enrichment. This finding suggested that melon sustainable production should control irrigation and adequately increase nitrogen fertilizer to trade−off fruit, quality, and efficiency for the expected future higher CO 2 concentrations. [Display omitted] • Fuzzy−Borda model was used to balance yield, quality, efficiency of melon. • Enriched CO 2 improved yield by 10.3 − 22.9%. • Enriched CO 2 improved free amino acid of low irrigation in all study seasons. • Intermediate irrigation and higher N input were best for melon under CO 2 enrichment. [ABSTRACT FROM AUTHOR]