自适应性可证明有帮助：非自适应感测的$l_0$成本的信息理论限制
Adaptivity provably helps: information-theoretic limits on $l_0$ cost of non-adaptive sensing
摘要：
自适应和反馈的优点在信号处理和通信中引起了极大的兴趣，有许多积极和消极的结果。虽然它是建立，自适应不提供大量减少极小极大均方误差为固定数量的测量，现有的结果已经显示出几个优点的自适应重建的复杂性，支持检测的准确性，和增益的信号-噪声比，在传感能量的约束下。感测能量通常根据感测矩阵的Frobenius范数来测量。本文使用一种不同的度量，我们称之为感知矩阵的$l_0$成本--来量化感知的复杂性。因此，稀疏传感矩阵具有较低的成本。我们推导出任何非自适应传感策略的$l_0$成本的信息理论下限。我们建立了任何非自适应传感策略必须招致的$l_0$成本$Θ\left( N \log_2(N)\right) $重建$N$维，一-稀疏信号时，测量的数量被限制为$Θ\left(\log_2 (N)\right)$。相比之下，对分型自适应策略对于相等数量的测量仅需要至多$\mathcal{O}(N)$的$l_0$成本。这个问题有一个有趣的解释，作为一个球包装问题在一个多维空间，使所有的球中心有最小的非零坐标。我们还讨论了$l_0$成本的变化作为测量的数量增加从$Θ\left(\log_2 (N)\right)$到$Θ\left(N\right)$。
英文摘要：
The advantages of adaptivity and feedback are of immense interest in signal processing and communication with many positive and negative results. Although it is established that adaptivity does not offer substantial reductions in minimax mean square error for a fixed number of measurements, existing results have shown several advantages of adaptivity in complexity of reconstruction, accuracy of support detection, and gain in signal-to-noise ratio, under constraints on sensing energy. Sensing energy has often been measured in terms of the Frobenius Norm of the sensing matrix. This paper uses a different metric that we call the $l_0$ cost of a sensing matrix-- to quantify the complexity of sensing. Thus sparse sensing matrices have a lower cost. We derive information-theoretic lower bounds on the $l_0$ cost that hold for any non-adaptive sensing strategy. We establish that any non-adaptive sensing strategy must incur an $l_0$ cost of $Θ\left( N \log_2(N)\right) $ to reconstruct an $N$-dimensional, one--sparse signal when the number of measurements are limited to $Θ\left(\log_2 (N)\right)$. In comparison, bisection-type adaptive strategies only require an $l_0$ cost of at most $\mathcal{O}(N)$ for an equal number of measurements. The problem has an interesting interpretation as a sphere packing problem in a multidimensional space, such that all the sphere centres have minimum non-zero co-ordinates. We also discuss the variation in $l_0$ cost as the number of measurements increase from $Θ\left(\log_2 (N)\right)$ to $Θ\left(N\right)$.
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