融合框架相位恢复
The Fusion Frame Phase Retrieval
- Huazhong University of Science and Technology（华中科技大学）
- Nankai University（南开大学）
- The University of Hong Kong（香港大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究融合框架相位恢复问题，利用秩r正交投影的集中不等式，证明梯度下降结合两阶段初始化以O(d log^2 d)测量复杂度实现线性收敛，并通过数值验证。
AI中文摘要：
相位恢复问题涉及仅从线性测量的幅度中重建函数或信号。大多数相位恢复算法的理论分析依赖于独立同分布的高斯随机测量或亚高斯随机测量。在本文中，我们关注融合框架相位恢复问题，其中采样矩阵是独立同分布的秩为$r$的正交投影，从Haar测度中抽取。我们给出了秩为$r$的正交投影矩阵集合上函数的集中不等式。这些不等式对于融合框架相位恢复问题的理论分析至关重要。基于这些不等式，我们证明了梯度下降结合两阶段初始化，在秩$r = O(1)$时，以$O(d\log^2 d)$的测量复杂度，实现到目标信号的线性收敛（直至全局相位）。我们通过数值结果验证了这一收敛性。
英文摘要：
The phase retrieval problem involves reconstructing a function or signal solely from the magnitude of linear measurements. Most theoretical analyses of phase retrieval algorithms rely on i.i.d. Gaussian random measurements or sub-Gaussian random measurements. In this paper, our focus is on the fusion frame phase retrieval problem, where the sampling matrices are i.i.d. rank-$r$ orthogonal projections drawn from the Haar measure. We present concentration inequalities for functions on the set of rank-$r$ orthogonal projection matrices. These inequalities are crucial for the theoretical analysis of the fusion frame phase retrieval problem. Based on these inequalities, we demonstrate that gradient descent, combined with a two-stage initialization, achieves linear convergence to the target signal up to a global phase with a measurement complexity of $O(d\log^2 d)$ when the rank $r = O(1)$. We verify this convergence through numerical results.
