Optimal partition recovery in general graphs

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.