We consider a graph-structured change point problem in which we observe a random vector with piecewise constant but unknown mean and whose independent, sub-Gaussian coordinates correspond to the $n$ nodes of a fixed graph. We are interested in the localisation task of recovering the partition of the nodes associated to the constancy regions of the mean vector. When the partition $\mathcal{S}$ consists of only two elements, we characterise the difficulty of the localisation problem in terms of four key parameters: the maximal noise variance $\sigma^2$, the size $\Delta$ of the smaller element of the partition, the magnitude $\kappa$ of the difference in the signal values across contiguous elements of the partition and the sum of the effective resistance edge weights $|\partial_r(\mathcal{S})|$ of the corresponding cut -- a graph theoretic quantity quantifying the size of the partition boundary. In particular, we demonstrate an information theoretical lower bound implying that, in the low signal-to-noise ratio regime $\kappa^2 \Delta \sigma^{-2} |\partial_r(\mathcal{S})|^{-1} \lesssim 1$, no consistent estimator of the true partition exists. On the other hand, when $\kappa^2 \Delta \sigma^{-2} |\partial_r(\mathcal{S})|^{-1} \gtrsim \zeta_n \log\{r(|E|)\}$, with $r(|E|)$ being the sum of effective resistance weighted edges and $\zeta_n$ being any diverging sequence in $n$, we show that a polynomial-time, approximate $\ell_0$-penalised least squared estimator delivers a localisation error -- measured by the symmetric difference between the true and estimated partition -- of order $ \kappa^{-2} \sigma^2 |\partial_r(\mathcal{S})| \log\{r(|E|)\}$. Aside from the $\log\{r(|E|)\}$ term, this rate is minimax optimal. Finally, we provide discussions on the localisation error for more general partitions of unknown sizes.
翻译:我们考虑一个图形结构化的更改点问题, 我们在这个问题上看到一个随机矢量, 其平面常数常数为0, 其独立的Gaussian 坐标与固定图形的美元节点相对应。 我们感兴趣的是恢复与平均矢量的凝固区域相关的节点分割的本地化任务。 当分区 $\ mathcal{S} 仅包含两个元素时, 我们用四个关键参数来描述本地化问题的难度: 最大噪音差异 $\ gma2, 分区较小元素的大小 $\ Delta美元, 分区毗连部分的信号值差异的大小 $ $\ kapta2; 直径平面的平面重量之和 $_\\\\\\\\\\ 美元; 平面平面的信号值是 ================================== 平面值, 我们显示, 低的信号比值制度 =======xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx