电视正则化压缩感知的恢复保证

C. Poon
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引用次数: 0

摘要

本文研究了从被噪声破坏的傅里叶系数的不完全子集中恢复梯度近似稀疏的一维或二维离散信号的问题。结果表明,为了获得对噪声具有鲁棒性和对s阶不精确梯度稀疏性具有高概率稳定性的重构,只需均匀随机抽取O(s log N)个可用的傅里叶系数即可。然而,如果根据集中于低傅立叶频率的特定分布绘制O(s log N)个样本,那么可以保证的稳定性界在对数因子范围内是最优的。本文的最终结果表明,在一维情况下,底层信号是梯度稀疏的,并且其稀疏模式满足最小分离条件,那么为了保证高概率精确恢复,对于M <;N,从频率不大于M的傅里叶系数中均匀随机抽取O(s log M)个样本就足够了。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Recovery guarantees for TV regularized compressed sensing
This paper considers the problem of recovering a one or two dimensional discrete signal which is approximately sparse in its gradient from an incomplete subset of its Fourier coefficients which have been corrupted with noise. The results show that in order to obtain a reconstruction which is robust to noise and stable to inexact gradient sparsity of order s with high probability, it suffices to draw O(s log N) of the available Fourier coefficients uniformly at random. However, if one draws O(s log N) samples in accordance to a particular distribution which concentrates on the low Fourier frequencies, then the stability bounds which can be guaranteed are optimal up to log factors. The final result of this paper shows that in the one dimensional case where the underlying signal is gradient sparse and its sparsity pattern satisfies a minimum separation condition, then to guarantee exact recovery with high probability, for some M <; N, it suffices to draw O(s log M logs) samples uniformly at random from the Fourier coefficients whose frequencies are no greater than M.
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