相对观测线性最优估计的误差缩放定律
Error Scaling Laws for Linear Optimal Estimation from Relative Measurements
摘要：
我们研究的问题，估计向量值的变量从嘈杂的“相对”的测量。这个问题出现在几个传感器网络应用中。测量模型可以用图来表示，图的节点对应于变量，而边对应于两个变量之间的差的噪声测量。我们以一个任意变量作为参考，并考虑剩余变量的最优（最小方差）线性无偏估计。我们研究了节点变量的最优线性无偏估计的误差如何随着节点到参考节点的距离而增长。我们建立了一个图的分类，即，稠密或稀疏的Rd，1<= d <=3，决定了一个节点的线性无偏最优估计误差如何增长与它的距离从参考节点。特别地，如果图在1、2或3D中是稠密的，则节点变量的估计误差分别由与参考的距离的线性、对数或有界函数上界。如果图在1、2和3D中是稀疏的，则得到相应的下界。我们的研究结果还表明，天真的措施，如节点度图密度，是不充分的预测估计误差。对于最优线性无偏估计，这些缩放定律确定了在大型图中可实现的估计精度的算法独立限制。
英文摘要：
We study the problem of estimating vector-valued variables from noisy "relative" measurements. This problem arises in several sensor network applications. The measurement model can be expressed in terms of a graph, whose nodes correspond to the variables and edges to noisy measurements of the difference between two variables. We take an arbitrary variable as the reference and consider the optimal (minimum variance) linear unbiased estimate of the remaining variables. We investigate how the error in the optimal linear unbiased estimate of a node variable grows with the distance of the node to the reference node. We establish a classification of graphs, namely, dense or sparse in Rd,1<= d <=3, that determines how the linear unbiased optimal estimation error of a node grows with its distance from the reference node. In particular, if a graph is dense in 1,2, or 3D, then a node variable's estimation error is upper bounded by a linear, logarithmic, or bounded function of distance from the reference, respectively. Corresponding lower bounds are obtained if the graph is sparse in 1, 2 and 3D. Our results also show that naive measures of graph density, such as node degree, are inadequate predictors of the estimation error. Being true for the optimal linear unbiased estimate, these scaling laws determine algorithm-independent limits on the estimation accuracy achievable in large graphs.
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