优化采样可有效降噪，理论证明测量越多误差越小
Denoising guarantees for optimized sampling schemes in compressed sensing
- 用优化采样代替均匀采样，提升压缩感知性能
- 高斯噪声下，测量数增加时重建误差趋近于零
- 适用于稀疏向量和生成网络输出等低维结构
压缩感知中使用子采样的酉矩阵时，优化采样方案相比均匀采样具有更优的理论保证与实证表现。本文首次在压缩感知领域提供理论证明：当噪声为高斯分布时，优化采样方案下测量噪声引起的误差随测量次数增加而趋于零。我们还对有放回且任意概率权重的采样给出类似保证。所有结果均适用于包含于若干低维子空间并集中的先验集合。实验表明，当先验为生成式ReLU神经网络的输出或稀疏向量时，该去噪行为在实际中以接近理论预测的速率出现。
原文摘要 · Abstract (English)
Compressed sensing with subsampled unitary matrices benefits from \emph{optimized} sampling schemes, which feature improved theoretical guarantees and empirical performance relative to uniform subsampling. We provide, in a first of its kind in compressed sensing, theoretical guarantees showing that the error caused by the measurement noise vanishes with an increasing number of measurements for optimized sampling schemes, assuming that the noise is Gaussian. We moreover provide similar guarantees for measurements sampled with-replacement with arbitrary probability weights. All our results hold on prior sets contained in a union of low-dimensional subspaces. Finally, we demonstrate that this denoising behavior appears in empirical experiments with a rate that closely matches our theoretical guarantees when the prior set is the range of a generative ReLU neural network and when it is the set of sparse vectors.
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