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 (with a good initialization) that produces a sequence linearly converging to $\boldsymbol{x}$ with $m=O(k\log n)$ measurements. Second, we explore the generative prior, assuming that $\boldsymbol{x}$ lies in the range of an $L$-Lipschitz continuous generative model with $k$-dimensional inputs in an $\ell_2$-ball of radius $r$. 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：问题：从二次系统 $\{y_i=\boldsymbol{x}^\top\boldsymbol{A}_i\boldsymbol{x},\ i=1,\ldots,m\}$（其中 $\boldsymbol{A}_i$ 为满秩矩阵）中恢复信号 $\boldsymbol{x} \in \mathbb{R}^n$ 在无赋值距离几何和亚波长成像等应用中频繁出现。本文针对 $\boldsymbol{A}_i$ 为独立同分布标准高斯矩阵的情形，通过引入 $\boldsymbol{x}$ 的先验知识，解决 $m\ll n$ 的高维问题。首先，我们考虑 $k$-稀疏的 $\boldsymbol{x}$，并提出无需稀疏度 $k$ 的阈值化 Wirtinger 流算法。该算法包含两步：当 $m=O(k^2\log n)$ 时，谱初始化能够识别出充分接近 $\boldsymbol{x}$ 的点（允许符号翻转）；当 $m=O(k\log n)$ 次测量时，阈值化梯度下降（配合良好的初始化）能产生线性收敛到 $\boldsymbol{x}$ 的序列。其次，我们探索生成先验，假设 $\boldsymbol{x}$ 位于一个 $L$-利普希茨连续生成模型的像集内，该模型输入为 $k$ 维且位于半径为 $r$ 的 $\ell_2$ 球中。我们提出投影梯度下降算法，同样包含两步：当 $m=O(k\log(Lnr))$ 次测量时，投影幂法提供初始向量，其 $\ell_2$ 误差为 $O\big(\sqrt{\frac{k \log L}{m}}\big)$；当 $m=O(k\log\frac{Lrn}{\delta^2})$ 时，投影梯度下降以几何速率将 $\ell_2$ 误差精化至 $O(\delta)$。实验结果验证了我们的理论发现，并表明：(i) 针对稀疏情形的方法显著优于现有可证明算法——稀疏幂分解；(ii) 利用生成先验能够从少量二次测量中精确恢复 MNIST 数据集中的图像。