二次可行性问题的样本复杂性与优化景观
On the Sample Complexity and Optimization Landscape for Quadratic Feasibility Problems
摘要：
我们考虑从$m$二次测量值$\{\langle A_i\mathbf{x}, \mathbf{x}\rangle\}_{i=1}^m$恢复复向量$\mathbf{x}\in \mathbb{C}^n$的问题。这个问题，被称为二次可行性，包括众所周知的相位恢复问题，并在广泛的重要领域，包括电力系统状态估计和X射线晶体学的应用。一般来说，二次可行性问题不仅NP-难解决，而且实际上可能无法识别。在本文中，我们建立了条件下，这个问题成为{可识别}，并进一步证明等距性质的情况下，矩阵$\{A_i\}_{i=1}^m$是厄米特矩阵采样的复高斯分布。此外，我们探讨了这个问题的非凸{优化}制定，并建立相关的优化景观，使梯度算法与任意初始化收敛到一个\emph{globally optimal}点的概率很高的显着特点。我们的研究结果还揭示了在这些情况下成功识别可行解决方案的样本复杂性要求。
英文摘要：
We consider the problem of recovering a complex vector $\mathbf{x}\in \mathbb{C}^n$ from $m$ quadratic measurements $\{\langle A_i\mathbf{x}, \mathbf{x}\rangle\}_{i=1}^m$. This problem, known as quadratic feasibility, encompasses the well known phase retrieval problem and has applications in a wide range of important areas including power system state estimation and x-ray crystallography. In general, not only is the the quadratic feasibility problem NP-hard to solve, but it may in fact be unidentifiable. In this paper, we establish conditions under which this problem becomes {identifiable}, and further prove isometry properties in the case when the matrices $\{A_i\}_{i=1}^m$ are Hermitian matrices sampled from a complex Gaussian distribution. Moreover, we explore a nonconvex {optimization} formulation of this problem, and establish salient features of the associated optimization landscape that enables gradient algorithms with an arbitrary initialization to converge to a \emph{globally optimal} point with a high probability. Our results also reveal sample complexity requirements for successfully identifying a feasible solution in these contexts.
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