In the anisotropic random geometric graph model, vertices correspond to points drawn from a high-dimensional Gaussian distribution and two vertices are connected if their distance is smaller than a specified threshold. We study when it is possible to hypothesis test between such a graph and an Erd\H{o}s-R\'enyi graph with the same edge probability. If $n$ is the number of vertices and $\alpha$ is the vector of eigenvalues, Eldan and Mikulincer show that detection is possible when $n^3 \gg (\|\alpha\|_2/\|\alpha\|_3)^6$ and impossible when $n^3 \ll (\|\alpha\|_2/\|\alpha\|_4)^4$. We show detection is impossible when $n^3 \ll (\|\alpha\|_2/\|\alpha\|_3)^6$, closing this gap and affirmatively resolving the conjecture of Eldan and Mikulincer.
翻译：在厌食性随机几何图形模型中,脊椎与从高斯高斯高位分布中抽取的点数相对应,如果距离小于规定的阈值,则两个脊椎相联。当有可能用同样的边缘概率对此类图表和厄尔德/H{o}s-R\'enyi图进行假设测试时,我们进行研究。如果美元是脊椎的数量,美元=阿尔法值的矢量,Eldan和Mikulincer显示,当美元=3\gg( ⁇ alpha ⁇ 2/ ⁇ alpha}3)时,发现是可能的,当美元=3\ll( ⁇ alpha2/ ⁇ alpha}4}4美元)时,则无法检测到。当美元=3时,我们显示无法检测到( ⁇ / ⁇ 2/ ⁇ alpha}3}6美元,缩小这一差距,肯定地解决Eldan和Mkulincer的等。
相关内容
微信扫码咨询专知VIP会员