We present a fully dynamic algorithm for correlation clustering that supports \emph{vertex} insertions and deletions while maintaining an $O(1)$ approximation in polylogarithmic amortized update time. This resolves an open problem of vertex deletions posed by a recent line of work on dynamic correlation clustering. We adopt the min-disagreement objective for correlation clustering, i.e., given a signed complete graph $G=(V,E^+\cup E^-)$, we want to minimize the total number of $(+)$ edge crossing clusters and $(-)$ edges within clusters. The input is given in the adjacency list model, where each update inserts or deletes a node along with its full set of $(+)$ neighbors. Our main technical contribution is a fault-tolerant sparse-dense decomposition framework which unlike the existing decomposition-based frameworks allows for structural inconsistencies as long as the induced cost is bounded. We also show a separation between explicitly maintaining the (stronger) sparse-dense decomposition and maintaining a constant-factor approximation to correlation clustering, which may be of independent interest.
Modern data systems often require aggregating information from multiple sources: some might be entirely untrustworthy, while the remaining presumably reliable sources may still suffer from a small amount of corruption. We study robust mean estimation under this practical scenario, which is formulated through the double corruption model: samples are divided among $N$ users, each with $n$ samples. $\varepsilon$ fraction of the users could be arbitrarily corrupted, while each of the remaining ``good'' users still have $α$-fraction ($α>0$) of corrupted samples. The goal is to estimate the unknown mean $μ$ with small $\ell_2$ error. This is a challenging scenario because we must handle two types of adversarial corruption: user-level corruption and sample-level corruption. When the unknown distribution has covariance matrix bounded by the identity matrix, we design a sum-of-squares (SoS) algorithm that achieves an $\ell_2$ error of $\sqrtα$ with probability at least $2/3$ as long as the total number of samples $nN=\widetildeΩ(d/α)$. The estimation error and sample complexity are min-max optimal under the mild condition. For comparison, treating all $nN$ samples as $(\varepsilon+α)$-corrupted and applying standard robust mean estimation only achieves an error of $\sqrt{\varepsilon+α}$. Thus, our new algorithm removes the adverse effect of untrustworthy users and achieves the optimal error as if only sample-level corruption is present. Also, our SoS algorithm extends to the heterogeneous setting where each user's local mean could be additionally shifted by $\sqrtβ$ in $\ell_2$ distance. We show that an error of $O(\sqrt{α+β})$ can be achieved with probability at least $2/3$ when $nN=\widetildeΩ(d/(α+β))$.