Many driving factors of physical systems are latent or unobserved. Thus, understanding such systems and producing robust predictions crucially relies on accounting for the influence of the latent structure. I will discuss methodological and theoretical advances in two important problems in latent-variable modeling. The first problem focuses on developing false discovery methods for latent-variable models that are parameterized by low-rank matrices, where the traditional perspective on false discovery control is ill-suited due to the non-discrete nature of the underlying decision spaces. To overcome this challenge, I will present a geometric reformulation of the notion of a discovery as well as a specific algorithm to control false discoveries in these settings. The second problem aims to estimate causal relations among a collection of observed variables with latent effects. Given access to data arising from perturbations (interventions), I will introduce a regularized maximum-likelihood framework that provably identifies the underlying causal structure and improves robustness to distributional changes. Throughout, I will explore the utility of the proposed methodologies for real-world applications such as water resource management.