Recovering Structured Data From Superimposed Non-Linear Measurements
Recovering Structured Data From Superimposed Non-Linear Measurements
复制标题
从叠加的非线性测量中恢复结构化数据
DOI：
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发表时间：
2017
影响因子：
2.5
通讯作者：
P. Jung
中科院分区：
文献类型：
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作者：
Martin Genzel;P. Jung
本文研究了非线性通信约束下的分布式数据采集问题。更具体地说，我们考虑一个模型设置，其中<inline-formula><tex-math notation="LaTeX">$M$</tex-math></inline-formula><italic>分布式节点</italic>采取一个未知的<italic>结构化源向量</italic><inline-formula>的单独测量<tex-math notation="LaTeX">。 oldsymbol {x}_{0}在mathbb {R}^{n}$中</tex-math></inline-formula>，将它们的读数同时传送到<italic>中央接收器</italic>。由于该过程涉及冲突并且通常是不完美的，所以接收器测量<italic>非线性失真信号的叠加</italic>。在第一步中，我们将证明一个<inline-formula><tex-math notation="LaTeX">$s$</tex-math></inline-formula>-稀疏向量<inline-formula><tex-math notation="LaTeX">$ oldsymbol {x}_{0}$</tex-math></inline-formula>可以使用传统的Lasso估计器从<inline-formula>这种叠加测量的<tex-math notation="LaTeX">$ O（s cdot log（2n/s）$</tex-math></inline-formula>中成功恢复，该估计器不依赖于关于非线性损坏的任何知识。然而，这种<italic>直接</italic>方法不能用于几种“未校准”的系统配置。这些<italic>盲重建任务</italic>可以通过<inline-formula><tex-math notation="LaTeX">$ ell ^{}$</tex-math></inline-formula>-Group-Lasso轻松处理，但沿着增加的采样率为<inline-formula><tex-math notation="LaTeX">$ O（scdot max {M，log（2n/s）}$</tex-math></inline-formula>观测值-实际上，这种<italic>提升</italic>策略的目的是将某类<italic>双线性逆问题</italic>扩展到<italic>非线性</italic>采集。我们的两个算法方法是一个更抽象的框架，其中包括亚高斯测量设计以及一般（凸）结构约束的一个特殊实例。这些结果对于各种恢复和学习任务具有独立的意义，因为它们适用于任意非线性观测模型。最后，为了说明我们的理论研究结果的实际范围，<italic>无线传感器网络</italic>的应用进行了讨论，这实际上是我们的方法的原型例子。
This work deals with the problem of distributed data acquisition under non-linear communication constraints. More specifically, we consider a model setup where <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> <italic>distributed nodes</italic> take individual measurements of an unknown <italic>structured source vector</italic> <inline-formula> <tex-math notation="LaTeX">$ oldsymbol {x}_{0}in mathbb {R}^{n}$ </tex-math></inline-formula>, communicating their readings simultaneously to a <italic>central receiver</italic>. Since this procedure involves collisions and is usually imperfect, the receiver measures a <italic>superposition of non-linearly distorted signals</italic>. In a first step, we will show that an <inline-formula> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula>-sparse vector <inline-formula> <tex-math notation="LaTeX">$ oldsymbol {x}_{0}$ </tex-math></inline-formula> can be successfully recovered from <inline-formula> <tex-math notation="LaTeX">$ O(s cdot log (2n/s)$ </tex-math></inline-formula> of such superimposed measurements, using a traditional Lasso estimator that does not rely on any knowledge about the non-linear corruptions. This <italic>direct</italic> method however fails to work for several “uncalibrated” system configurations. These <italic>blind reconstruction tasks</italic> can be easily handled with the <inline-formula> <tex-math notation="LaTeX">$ ell ^{}$ </tex-math></inline-formula>-Group-Lasso, but coming along with an increased sampling rate of <inline-formula> <tex-math notation="LaTeX">$ O(scdot max {M, log (2n/s) }$ </tex-math></inline-formula> observations — in fact, the purpose of this <italic>lifting</italic> strategy is to extend a certain class of <italic>bilinear inverse problems</italic> to <italic>non-linear</italic> acquisition. Our two algorithmic approaches are a special instance of a more abstract framework which includes sub-Gaussian measurement designs as well as general (convex) structural constraints. These results are of independent interest for various recovery and learning tasks, as they apply to arbitrary non-linear observation models. Finally, to illustrate the practical scope of our theoretical findings, an application to <italic>wireless sensor networks</italic> is discussed, which actually serves as the prototypical example of our methodology.
影响因子：
1.6
作者：
Amelunxen, Dennis;Lotz, Martin;Tropp, Joel A.
通讯作者：
Tropp, Joel A.