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Matrix completion tackles the task of predicting missing values in a low-rank matrix based on a sparse set of observed entries. It is often assumed that the observation pattern is generated uniformly at random or has a very specific structure tuned to a given algorithm. There is still a gap in our understanding when it comes to arbitrary sampling patterns. Given an arbitrary sampling pattern, we introduce a matrix completion algorithm based on network flows in the bipartite graph induced by the observation pattern. For additive matrices, we show that the electrical flow is optimal, and we establish error upper bounds customized to each entry as a function of the observation set, along with matching minimax lower bounds. Our results show that the minimax squared error for recovery of a particular entry in the matrix is proportional to the effective resistance of the corresponding edge in the graph. Furthermore, we show that the electrical flow estimator is equivalent to the least squares estimator. We apply our estimator to the two-way fixed effects model and show that it enables us to accurately infer individual causal effects and the unit-specific and time-specific confounders. For rank-1 matrices, we use edge-disjoint paths to form an estimator that achieves minimax optimal estimation when the sampling is sufficiently dense. Our discovery introduces a new family of estimators parametrized by network flows, which provide a fine-grained and intuitive understanding of the impact of the given sampling pattern on the difficulty of estimation at an entry-specific level. This graph-based approach allows us to quantify the inherent complexity of matrix completion for individual entries, rather than relying solely on global measures of performance. 

We report the in operando visualization of the photocatalytic turnovers on single eosin Y (EY) through a redox-induced photoblinking phenomenon. The photocatalytic cyclization of thiobenzamide (TB) catalyzed by EY was investigated. The analysis of the intensity-versus-time trajectories of single EYs revealed the kinetics and dynamics of the elementary photocatalytic turnovers and the heterogeneity of the activity of individual EYs. The quenching turnover time showed a fast population and a slow population, which could be attributed to the singlet and triplet states of photoexcited EY. The slow quenching turnovers were more dominant at higher TB concentrations. The activity heterogeneity of EYs was studied over a series of reactant concentrations. Excess quenching reagent was found to decrease the percentage of active EYs. The method can be broadly applied to studying the elementary processes of photocatalytic organic reactions in operando.