Quantized compressed sensing for random circulant matrices
Quantized compressed sensing for random circulant matrices
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DOI：
10.1016/j.acha.2019.03.004
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发表时间：
2019-03
影响因子：
2.5
通讯作者：
Joe-Mei Feng;F. Krahmer;Rayan Saab
中科院分区：
文献类型：
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作者：
Joe-Mei Feng;F. Krahmer;Rayan Saab
我们首次分析了一种用于结构化测量的压缩感知的非平凡量化方案。我们考虑由随机次高斯向量生成的循环矩阵中随机选择的行组成的压缩传感矩阵。我们使用稳定的，可能是一位的Sigma-Delta方案对测量结果进行量化，并通过凸优化来重建信号。我们证明了由量化引起的重建误差部分随着测量次数的增加而呈多项式衰减。这与随机亚高斯矩阵的Sigma-Delta量化的类似结果是一致的，并且明显好于广泛假设的无记忆标量量化的结果。此外，我们证明了我们的方法是稳定的和稳健的；在存在噪声和潜在信号不是严格稀疏的情况下，重建误差优雅地下降。该分析依赖于有关次高斯型混沌过程的结果和麦克迪尔米德不等式的变体。
We provide the first analysis of a non-trivial quantization scheme for compressed sensing with structured measurements. We consider compressed sensing matrices consisting of rows selected randomly, without replacement, from a circulant matrix generated by a random subgaussian vector. We quantize the measurements using stable, possibly one-bit, Sigma-Delta schemes, and reconstruct the signal via convex optimization. We show that the part of the reconstruction error due to quantization decays polynomially in the number of measurements. This is in-line with analogous results on Sigma-Delta quantization associated with random subgaussian matrices, and significantly better than results associated with the widely assumed memoryless scalar quantization. Moreover, we prove our approach is stable and robust; the reconstruction error degrades gracefully in the presence of noise and when the underlying signal is not strictly sparse. The analysis relies on results concerning subgaussian chaos processes and a variation of McDiarmid's inequality.