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首页> 外文期刊>IEEE Transactions on Information Theory >Living Near the Edge: A Lower-Bound on the Phase Transition of Total Variation Minimization
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Living Near the Edge: A Lower-Bound on the Phase Transition of Total Variation Minimization

机译：生活在边缘附近：较低的阶段转换总变化最小化

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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.
机译：这项工作是关于最小化的总变化（电视）最小化，用于从压缩测量中恢复梯度稀疏信号。最近的研究表明，随着测量次数增加，电视最小化表现出从未成功的相位过渡行为。实际上，在大尺寸方面，电视最小化成功地在测量次数超过某个阈值时恢复具有高概率的梯度 - 稀疏信号;否则，它几乎肯定会失败。获取近似该阈值的闭合表格表达式是该字段中的主要挑战，并且尚未适当地解决。在这项工作中，在任何随机测量矩阵的情况下，我们在该阈值下得出了紧密的较低限制，其空空格被均匀地分布到哈尔测量。与依赖于简单梯度稀疏度水平的传统电视相转变结果相比，我们的约束受梯度稀疏性的广义概念的影响。我们提出的绑定非常接近通过模拟结果证实的电视最小化的真实相变。

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