This work deals with the problem of distributed data acquisition under non-linear communication constraints. More specifically, we consider a model setup where $M$ distributed nodes take individual measurements of an unknown structured source vector $ \boldsymbol {x}_{0}\in \mathbb {R}^{n}$ , communicating their readings simultaneously to a central receiver. Since this procedure involves collisions and is usually imperfect, the receiver measures a superposition of non-linearly distorted signals. In a first step, we will show that an $s$ -sparse vector $ \boldsymbol {x}_{0}$ can be successfully recovered from $ O(s \cdot \log (2n/s)$ of such superimposed measurements, using a traditional Lasso estimator that does not rely on any knowledge about the non-linear corruptions. This direct method however fails to work for several “uncalibrated” system configurations. These blind reconstruction tasks can be easily handled with the $ \ell ^{}$ -Group-Lasso, but coming along with an increased sampling rate of $ O(s\cdot \max \{M, \log (2n/s) \}$ observations — in fact, the purpose of this lifting strategy is to extend a certain class of bilinear inverse problems to non-linear 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 wireless sensor networks is discussed, which actually serves as the prototypical example of our methodology.

中文翻译：

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从叠加的非线性测量中恢复结构化数据

这项工作处理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 百万美元 分布式节点 对未知进行单独测量 结构化源向量 $ \boldsymbol {x}_{0}\in \mathbb {R}^{n}$ ，同时将他们的读数传达给一个 中央接收器. 由于这个过程涉及碰撞并且通常是不完美的，接收器测量一个非线性失真信号的叠加. 在第一步中，我们将展示一个 $s$ -稀疏向量 $ \boldsymbol {x}_{0}$ 可以成功恢复 $ O(s \cdot \log (2n/s)$ 使用不依赖任何关于非线性损坏的知识的传统套索估计器对此类叠加测量进行分析。这直接的然而，该方法不适用于几种“未校准”的系统配置。这些盲重建任务 可以很容易地处理 $ \ell ^{}$ -Group-Lasso，但随着采样率的增加而增加 $ O(s\cdot \max \{M, \log (2n/s) \}$ 观察——事实上，这样做的目的 举重 策略是扩展某一类 双线性逆问题 至 非线性获得。我们的两种算法方法是更抽象框架的一个特殊实例，其中包括亚高斯测量设计以及一般（凸）结构约束。这些结果对于各种恢复和学习任务具有独立意义，因为它们适用于任意非线性观察模型。最后，为了说明我们的理论发现的实际范围，应用到无线传感器网络 讨论，这实际上是我们方法论的典型例子。