Asymptotic Convergence Rates of the Length of the Longest Run(s) in an Inflating Bernoulli Net

Ni Kai; Cao Shanshan; Huo Xiaoming

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Asymptotic Convergence Rates of the Length of the Longest Run(s) in an Inflating Bernoulli Net

机译：膨胀伯努利网中最长次运行长度的渐近收敛率

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摘要

In image detection, one problem is to test whether the set, though mainly consisting of uniformly scattered points, also contains a small fraction of points sampled from some (a priori unknown) curve, for example, a curve with C-alpha-norm bounded by beta. One approach is to analyze the data by counting membership in multiscale multianisotropic strips, which involves an algorithm that delves into the length of the path connecting many consecutive "significant" nodes. In this paper, we develop the mathematical formalism of this algorithm and analyze the statistical property of the length of the longest significant run. The rate of convergence is derived. Using percolation theory and random graph theory, we present a novel probabilistic model named, pseudo-tree model. Based on the asymptotic results for the pseudo-tree model, we further study the length of the longest significant run in an "inflating" Bernoulli net. We find that the probability parameter p of significant node plays an important role: there is a threshold pc, such that in the cases of p < p(c) and p > p(c), very different asymptotic behaviors of the length of the significant runs are observed. We apply our results to the detection of an underlying curvilinear feature and prove that the test based on our proposed longest run theory is asymptotically powerful.

机译：在图像检测中，一个问题是测试该集合是否主要由均匀散射的点组成，还包含从一些（先验未知）曲线上采样的小部分，例如，具有C-alpha-Norm的曲线通过β。一种方法是通过计算多尺度的多色点中的成员资格来分析数据，这涉及一种算法，该算法删除了连接许多连续的“重要”节点的路径的长度。在本文中，我们开发了该算法的数学形式主义，并分析了最长持续运行长度的统计特性。收敛速度是推导的。使用渗滤理论和随机图理论，我们提出了一种名为伪树模型的新型概率模型。基于伪树模型的渐近结果，我们进一步研究了“充气”Bernoulli网中最长的重大运行的长度。我们发现显着节点的概率参数P起着重要作用：存在阈值PC，使得在P p（c）的情况下，非常不同的渐近行为的长度观察到的跑步。我们将结果应用于检测底层曲线特征，并证明基于我们所提出的最长的运行理论的测试是渐近强大的。

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来源
《IEEE Transactions on Information Theory》 |2021年第9期|5922-5941|共20页
作者
Ni Kai; Cao Shanshan; Huo Xiaoming;
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作者单位

Georgia Inst Technol Coll Comp Atlanta GA 30332 USA;

Eli Lilly & Co Lilly Corp Ctr Indianapolis IN 46285 USA;

Georgia Inst Technol H Milton Stewart Sch Ind & Syst Engn Atlanta GA 30332 USA;

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正文语种 eng
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Testing; Target tracking; Streaming media; Noise measurement; Lattices; Convergence; Shape; Inflating Bernoulli net; pseudo-tree model; longest significant run; curve detection; asymptotically powerful test;

机译：测试;目标跟踪;流媒体;噪声测量;格子;融合;形状;膨胀Bernoulli网;伪树模型;伪树模型;最长的重大运行;曲线检测;曲线检测;曲线检测;曲线检测;曲线检测;

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Asymptotic Convergence Rates of the Length of the Longest Run(s) in an Inflating Bernoulli Net

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