We consider a novel graph-structured change point problem. We observe a random vector with piecewise constant mean and whose independent, sub-Gaussian coordinates correspond to the $n$ nodes of a fixed graph. We are interested in recovering the partition of the nodes associated to the constancy regions of the mean vector. Although graph-valued signals of this type have been previously studied in the literature for the different tasks of testing for the presence of an anomalous cluster and of estimating the mean vector, no localisation results are known outside the classical case of chain graphs. When the partition $\mathcal{S}$ consists of only two elements, we characterise the difficulty of the localisation problem in terms of: 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 and the sum of the effective resistance edge weights $|\partial_r(\mathcal{S})|$ of the corresponding cut. 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 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 upper bounds on the localisation error for more general partitions of unknown sizes.

翻译：我们考虑的是新颖的图形结构变化点问题。 我们观察到一个随机矢量, 其字符串恒定平均值, 其独立的 Gausian 坐标与固定图形的美元节点相对应 。 我们有兴趣恢复与平均矢量的粘结区域相关的节点分隔 。 虽然文献中已经研究过这种类型的图形值信号, 用于测试异常组的存在和估计平均矢量的不同任务, 但是在链条图的经典案例之外, 没有已知的本地化结果 。 当分区 $\ macal{S} 由两个元素组成时, 我们描述本地化问题的难度 : 最大噪音差异 $\ gma2 $, 小部分的 美元值, 信号值的值 $\ kapppa 和有效阻力的数值 $\\\\ \\\ \\\ \ \ \ \ \ \ \\ \ \ \ \ \ \ \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \