Explore advanced techniques for making accurate inferences about complex distributions when sample sizes are insufficient for empirical distributions to be reliable. Delve into three key problems: optimally de-noising empirical distributions to improve accuracy, estimating population spectra from limited high-dimensional data, and recovering low-rank approximations of probability matrices from observed count data. Learn about an instance-optimal learning algorithm for distribution approximation, methods for estimating unseen elements in larger samples, and applications to genomics. Examine approaches for accurately estimating covariance matrix eigenvalues in high-dimensional settings with limited samples. Investigate techniques for matrix recovery problems related to community detection and word embeddings. Gain insights from cutting-edge research on overcoming data limitations in statistical inference and machine learning applications.