The last decade of John Cozzens's tenure at the NSF witnessed the advent of theory and methods at the heart of modern data science. These advances include (but are not limited to) compressed sensing, sparse coding, inference methods robust to outliers and missing data, and convex optimization tools that facilitate a host of novel inference methods. This paper describes how these methods evolved from classical basis representations of signals to alternative, flexible representations of signal structure. These new representations facilitate more accurate and robust inference in many contexts, and research at the intersection of signal processing, machine learning, and optimization make it possible to learn new representations from complex sensor data. This paper explores several key representations that have emerged in the past decade and their impact on the signal processing community.