生活在边缘附近：全变差最小化相变的下限
Living near the edge: A lower-bound on the phase transition of total variation minimization
摘要：
本文研究了从压缩测量数据中恢复梯度稀疏信号的全变差最小化方法。最近的研究表明，电视最小化表现出从失败到成功的相变行为，随着测量次数的增加。事实上，在大尺寸，TV最小化成功地恢复梯度稀疏信号的概率很高时，测量的数量超过一定的阈值，否则，它几乎肯定会失败。获得近似于该阈值的封闭形式的表达式是该领域中的主要挑战，并且尚未得到适当的解决。在这项工作中，我们推导出一个紧的下限在这个阈值的情况下，任何随机测量矩阵的零空间是均匀分布的Haar措施。与依赖于简单的梯度稀疏水平的常规TV相变结果相反，我们的界受到广义梯度稀疏概念的高度影响。我们提出的界限是非常接近的真实相变的电视最小化证实了模拟结果。
英文摘要：
This work is about the total variation (TV) minimization which is used for recovering gradient-sparse signals from compressed measurements. Recent studies indicate that TV minimization exhibits a phase transition behavior from failure to success as the number of measurements increases. In fact, in large dimensions, TV minimization succeeds in recovering the gradient-sparse signal with high probability when the number of measurements exceeds a certain threshold; otherwise, it fails almost certainly. Obtaining a closed-form expression that approximates this threshold is a major challenge in this field and has not been appropriately addressed yet. In this work, we derive a tight lower-bound on this threshold in case of any random measurement matrix whose null space is distributed uniformly with respect to the Haar measure. In contrast to the conventional TV phase transition results that depend on the simple gradient-sparsity level, our bound is highly affected by generalized notions of gradient-sparsity. Our proposed bound is very close to the true phase transition of TV minimization confirmed by simulation results.
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