论文标题
HADAMARD WIRWINGER流量以稀疏相检索
Hadamard Wirtinger Flow for Sparse Phase Retrieval
论文作者
论文摘要
我们考虑了从一组无噪声级的测量值重建$ n $二维$ k $ -sparse信号的问题。将问题提出为不规则的经验风险最小化任务，我们研究了使用Hadamard参数化梯度下降的样本复杂性表现，我们称之为Hadamard Wirewtinger流量（HWF）。提供了对信号稀疏$ k $的知识，我们证明了HWF的一个步骤能够从$ k（x^*_ {max}）^{ - 2} $（modulo boogarithmic术语）中恢复支持，其中$ x^*_ {max _ {max} $是信号的最大组成部分。该支持恢复过程可用于初始化现有的重建方法，并产生与读取数据成本成比例和改善样品复杂性成本成正比的算法，当信号中包含至少一个大组件时，该算法在$ k $中是线性的。我们从数值上研究了HWF在收敛时的性能，并表明，尽管不需要任何明确的正规化形式，也不需要$ K $的知识，但HWF适应信号稀疏性，并且比现有基于梯度的方法更少的测量值重建稀疏信号。
We consider the problem of reconstructing an $n$-dimensional $k$-sparse signal from a set of noiseless magnitude-only measurements. Formulating the problem as an unregularized empirical risk minimization task, we study the sample complexity performance of gradient descent with Hadamard parametrization, which we call Hadamard Wirtinger flow (HWF). Provided knowledge of the signal sparsity $k$, we prove that a single step of HWF is able to recover the support from $k(x^*_{max})^{-2}$ (modulo logarithmic term) samples, where $x^*_{max}$ is the largest component of the signal in magnitude. This support recovery procedure can be used to initialize existing reconstruction methods and yields algorithms with total runtime proportional to the cost of reading the data and improved sample complexity, which is linear in $k$ when the signal contains at least one large component. We numerically investigate the performance of HWF at convergence and show that, while not requiring any explicit form of regularization nor knowledge of $k$, HWF adapts to the signal sparsity and reconstructs sparse signals with fewer measurements than existing gradient based methods.
