A number of inference problems with sensor networks involve projecting a measured signal onto a given subspace. In existing decentralized approaches, sensors communicate with their local neighbors to obtain a sequence of iterates that asymptotically converges to the desired projection. In contrast, the present paper develops methods that produce these projections in a finite and approximately minimal number of iterations. Building upon tools from graph signal processing, the problem is cast as the design of a graph filter which, in turn, is reduced to the design of a suitable graph shift operator. Exploiting the eigenstructure of the projection and shift matrices leads to an objective whose minimization yields approximately minimum-order graph filters. To cope with the fact that this problem is not convex, the present work introduces a novel convex relaxation of the number of distinct eigenvalues of a matrix based on the nuclear norm of a Kronecker difference. To tackle the case where there exists no graph filter capable of implementing a certain subspace projection with a given network topology, a second optimization criterion is presented to approximate the desired projection while trading the number of iterations for approximation error. Two algorithms are proposed to optimize the aforementioned criteria based on the alternating-direction method of multipliers. An exhaustive simulation study demonstrates that the obtained filters can effectively obtain subspace projections markedly faster than existing algorithms.

中文翻译：

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用于分散子空间投影的快速图过滤器

传感器网络的许多推理问题涉及将测量信号投影到给定的子空间。在现有的分散式方法中，传感器与其本地邻居进行通信以获得渐近收敛到所需投影的迭代序列。相比之下，本论文开发了在有限且近似最少的迭代次数中产生这些投影的方法。基于图信号处理的工具，该问题被视为图滤波器的设计，而图滤波器又被简化为合适的图移位算子的设计。利用投影和移位矩阵的特征结构导致目标的最小化产生近似最小阶图滤波器。为了应对这个问题不是凸的，目前的工作基于 Kronecker 差异的核范数引入了矩阵的不同特征值数量的新凸松弛。为了解决不存在能够使用给定网络拓扑实现某个子空间投影的图过滤器的情况，提出了第二个优化标准来逼近所需的投影，同时用迭代次数换取逼近误差。基于乘法器的交替方向方法，提出了两种算法来优化上述标准。详尽的模拟研究表明，所获得的滤波器可以比现有算法明显更快地有效地获得子空间投影。为了解决不存在能够使用给定网络拓扑实现某个子空间投影的图过滤器的情况，提出了第二个优化标准来逼近所需的投影，同时用迭代次数换取逼近误差。基于乘法器的交替方向方法，提出了两种算法来优化上述标准。详尽的模拟研究表明，所获得的滤波器可以比现有算法明显更快地有效地获得子空间投影。为了解决不存在能够使用给定网络拓扑实现某个子空间投影的图过滤器的情况，提出了第二个优化标准来逼近所需的投影，同时用迭代次数换取逼近误差。基于乘法器的交替方向方法，提出了两种算法来优化上述标准。详尽的模拟研究表明，所获得的滤波器可以比现有算法明显更快地有效地获得子空间投影。基于乘法器的交替方向方法，提出了两种算法来优化上述标准。详尽的模拟研究表明，所获得的滤波器可以比现有算法明显更快地有效地获得子空间投影。基于乘法器的交替方向方法，提出了两种算法来优化上述标准。详尽的模拟研究表明，所获得的滤波器可以比现有算法明显更快地有效地获得子空间投影。