We study generative compressed sensing when the measurement matrix is randomly subsampled from a unitary matrix (with the DFT as an important special case). It was recently shown that $\textit{O}(kdn\| \boldsymbol{\alpha}\|_{\infty}^{2})$ uniformly random Fourier measurements are sufficient to recover signals in the range of a neural network $G:\mathbb{R}^k \to \mathbb{R}^n$ of depth $d$, where each component of the so-called local coherence vector $\boldsymbol{\alpha}$ quantifies the alignment of a corresponding Fourier vector with the range of $G$. We construct a model-adapted sampling strategy with an improved sample complexity of $\textit{O}(kd\| \boldsymbol{\alpha}\|_{2}^{2})$ measurements. This is enabled by: (1) new theoretical recovery guarantees that we develop for nonuniformly random sampling distributions and then (2) optimizing the sampling distribution to minimize the number of measurements needed for these guarantees. This development offers a sample complexity applicable to natural signal classes, which are often almost maximally coherent with low Fourier frequencies. Finally, we consider a surrogate sampling scheme, and validate its performance in recovery experiments using the CelebA dataset.

翻译：我们研究了当测量矩阵从酉矩阵（以离散傅里叶变换为重要特例）中随机子采样时，生成式压缩感知问题。最近的研究表明，利用$\textit{O}(kdn\| \boldsymbol{\alpha}\|_{\infty}^{2})$个均匀随机傅里叶测量即可恢复深度为$d$的神经网络$G:\mathbb{R}^k \to \mathbb{R}^n$值域内的信号，其中局部相干性向量$\boldsymbol{\alpha}$的每个分量量化了相应傅里叶向量与网络$G$值域的对齐程度。本文构建了一种模型自适应采样策略，其样本复杂度改进为$\textit{O}(kd\| \boldsymbol{\alpha}\|_{2}^{2})$个测量。该成果得益于：（1）我们为非均匀随机采样分布建立的新理论恢复保证；（2）优化采样分布以最小化满足这些保证所需的测量数。这一发展为适用于自然信号类的样本复杂度提供了理论依据——此类信号通常与低频傅里叶分量几乎达到最大相干性。最后，我们设计了一种替代采样方案，并通过CelebA数据集的恢复实验验证了其性能。