In this work, we delve into two statistical models: Gaussian mixture models and matrix completion, aiming to unravel intricate phenomena and develop effective algorithms with provable guarantees. First, our investigation discovers a nuanced phenomenon named Mode Melding in over-specified Gaussian mixture models, where we show that, under a covariance condition, the maximum likelihood estimator can achieve zero statistical error, in contrast to the slow non-parametric error rate usually present in over-specified models. By studying the landscape of the empirical log-likelihood function, we observe that the zero error corresponds to the case where the local maxima of the log-likelihood merge into one. This discovery not only helps us better understand over-specification but also inspires the design of our ReScaledGD optimization algorithm, highlighting the need for tailored methodologies in finite-sample scenarios. In the realm of low-rank matrix completion, we consider estimating unobserved entries in a low-rank matrix based on a sparse set of observed entries. Specifically, we study entry-specific guarantees to recover a low-rank matrix under highly non-uniform sampling, where the observed entries are sampled with highly varying probabilities, potentially with different asymptotic scalings. We demonstrate the efficacy of leveraging smaller but more densely observed submatrices under structured sampling probabilities, thereby advancing our understanding of fine-grained estimation difficulty through entry-specific error bounds dependent on local observation patterns. In additive models, we develop an estimator defined by the electrical flow in the bipartite graph generated by the sampling pattern, achieving minimax entry-specific error bounds proportional to the effective resistance, related to the connectivity in the underlying graph. This discovery not only deepens our understanding of additive models but also holds practical implications for real-world applications, particularly in domains requiring entry-specific characterization of estimation quality. Our results have applications in estimating individual causal effects with panel data assuming two-way fixed effects, illuminating the relative estimation difficulty as a function of the connectivity between vertices. Last but not least, we discuss an interesting application of matrix completion to the problem of offline Reinforcement Learning. We develop efficient offline algorithms by leveraging latent structures resulting in Q functions with a low-rank matrix representation. We introduce an offline policy evaluation algorithm that utilizes this low-rank characteristic to estimate values for state-action pairs that are not covered. Remarkably, our approach operates without needing a known feature representation, and our finite-sample error bound depends on a novel spectral discrepancy metric between the behavior and target policies. In addition, we demonstrate specific cases where our algorithm provides accurate estimates, even when traditional coverage conditions are not met.