The problem of recovering a signal $\boldsymbol x\in \mathbb{R}^n$ from a quadratic system $\{y_i=\boldsymbol x^\top\boldsymbol A_i\boldsymbol x,\ i=1,\ldots,m\}$ with full-rank matrices $\boldsymbol A_i$ frequently arises in applications such as unassigned distance geometry and sub-wavelength imaging. With i.i.d. standard Gaussian matrices $\boldsymbol A_i$, this paper addresses the high-dimensional case where $m\ll n$ by incorporating prior knowledge of $\boldsymbol x$. First, we consider a $k$-sparse $\boldsymbol x$ and introduce the thresholded Wirtinger flow (TWF) algorithm that does not require the sparsity level $k$. TWF comprises two steps: the spectral initialization that identifies a point sufficiently close to $\boldsymbol x$ (up to a sign flip) when $m=O(k^2\log n)$, and the thresholded gradient descent which, when provided a good initialization, produces a sequence linearly converging to $\boldsymbol x$ with $m=O(k\log n)$ measurements. Second, we explore the generative prior, assuming that $x$ lies in the range of an $L$-Lipschitz continuous generative model with $k$-dimensional inputs in an $\ell_2$-ball of radius $r$. With an estimate correlated with the signal, we develop the projected gradient descent (PGD) algorithm that also comprises two steps: the projected power method that provides an initial vector with $O\big(\sqrt{\frac{k \log L}{m}}\big)$ $\ell_2$-error given $m=O(k\log(Lnr))$ measurements, and the projected gradient descent that refines the $\ell_2$-error to $O(\delta)$ at a geometric rate when $m=O(k\log\frac{Lrn}{\delta^2})$. Experimental results corroborate our theoretical findings and show that: (i) our approach for the sparse case notably outperforms the existing provable algorithm sparse power factorization; (ii) leveraging the generative prior allows for precise image recovery in the MNIST dataset from a small number of quadratic measurements.

翻译：从二次系统 $\{y_i=\boldsymbol x^\top\boldsymbol A_i\boldsymbol x,\ i=1,\ldots,m\}$ 中恢复信号 $\boldsymbol x\in \mathbb{R}^n$ 的问题，在未分配距离几何学和亚波长成像等应用中经常出现，其中矩阵 $\boldsymbol A_i$ 为满秩矩阵。针对独立同分布的标准高斯矩阵 $\boldsymbol A_i$，本文通过引入 $\boldsymbol x$ 的先验知识，研究 $m\ll n$ 的高维情形。首先，我们考虑 $k$-稀疏的 $\boldsymbol x$，提出了无需已知稀疏度 $k$ 的阈值化Wirtinger流（TWF）算法。TWF包含两个步骤：当 $m=O(k^2\log n)$ 时，通过谱初始化找到一个充分接近 $\boldsymbol x$（允许符号翻转）的点；在获得良好初始化的前提下，当测量次数 $m=O(k\log n)$ 时，通过阈值化梯度下降产生线性收敛于 $\boldsymbol x$ 的序列。其次，我们探究生成先验，假设 $x$ 位于一个 $L$-Lipschitz连续生成模型的像集中，该模型的 $k$ 维输入位于半径为 $r$ 的 $\ell_2$ 球内。在获得与信号相关的估计后，我们提出了同样包含两个步骤的投影梯度下降（PGD）算法：当 $m=O(k\log(Lnr))$ 时，投影幂法提供的初始向量具有 $O\big(\sqrt{\frac{k \log L}{m}}\big)$ 的 $\ell_2$ 误差；当 $m=O(k\log\frac{Lrn}{\delta^2})$ 时，投影梯度下降以几何速率将 $\ell_2$ 误差优化至 $O(\delta)$。实验结果验证了我们的理论发现并表明：（i）针对稀疏情形的算法显著优于现有可证明算法稀疏功率分解；（ii）利用生成先验可从少量二次测量中精确恢复MNIST数据集中的图像。