确定最少观测次数是相位恢复、矩阵恢复中的一个基本问题。采用代数簇的方法，给出了通过无相位观测恢复几乎所有信号所需最少观测次数 [1]，并进一步将其扩展到通过线性观测恢复几乎所有低秩矩阵所需最少观测次数 [2]。对复稀疏信号的相位恢复问题，研究了L1模型的性能，成功将压缩感知中的主要结果扩展到无相位观测 [3]，并给出了求解稀疏信号相位恢复的S-PhaseLiftOff模型及求解算法 [4]。框架在相位恢复中扮演重要角色。其中一个基本问题是达到p-势能最小的框架位置，人们通过大量数值计算，猜想了其位置。我们证明了该猜想成立 [5]。
所提出的研究相位恢复的代数簇方法，促进了 Homomorphic Sensing 的发展，该理论研究重建信号的代数方法。解决了能否利用L1模型，能否通过O(k logd)次无相位观测恢复k-稀疏信号的开问题。对于p-势能最小化猜想的工作，文献中提到 “While revising this manuscript, we became aware of the work by Zhiqiang Xu and Zili Xu [27], where they prove the rest of the conjecture” （Advance Comp Math, 2020）; “the conjecture mentioned above from the paper [6] was proved true in [25].” (SIAM J. DISCRETE MATH., 2020)
1. Meng Huang, Yi Rong, Yang Wang and Zhiqiang Xu, Almost Everywhere Generalized Phase Retrieval, Applied and Computational Harmonic Analysis, Vol. 50, January 2021, Pages 16-33
2. Yi Rong, Yang Wang and Zhiqiang Xu, Almost everywhere injectivity conditions for the matrix recovery problem, Applied and Computational Harmonic Analysis, Vol.50 January 2021, Pages 386-400
3. Yu Xia and Zhiqiang Xu, The recovery of complex sparse signals from few phaseless measurements, Applied and Computational Harmonic Analysis, Volume 50,2021.
4. Yu Xia and Zhiqiang Xu, Sparse phase retrieval via Phaseliftoff, IEEE Transactions on Signal Processing, Page(s): 2129-2143, Vol.69, 18, 2021.
5. Zhiqiang Xu and Zili Xu, The minimizers of the p-frame potential, Applied and Computational Harmonic Analysis, Volume 52, May 2021, Pages 366-379
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