In a diverse set of problems ranging from dimensionality reduction to earthquake imaging it is often the case that we seek to identify sparse or quantized representations of signals. The rationale for this may be: 1) physical (I believe that there are only a few interesting things), 2) philosophical (I only want to think about a few interesting things), or 3) computational (I only have enough computer to work with a few interesting things). The past two decades have seen a radical progress in efficiently solving sparse recovery problems and these approaches are now being used for non-trivial estimation problems. This talk focuses on sparse and quantized signal recovery from historical, geometric, and philosophical perspectives, with examples from earthquake physics.