  Large diffuse galaxies are hard to find, but understanding the environments
where they live, their numbers, and ultimately their origins, is of intense
interest and importance for galaxy formation and evolution. Using Subaru's
Hyper Suprime-Cam Strategic Survey Program, we perform a systematic search for
low surface brightness galaxies and present novel and effective methods for
detecting and modeling them. As a case study, we surveyed 922 Milky Way analogs
in the nearby Universe ($0.01 < z < 0.04$) and build a large sample of
satellite galaxies that are outliers in the mass-size relation. These
``ultra-puffy'' galaxies (UPGs), defined to be $1.5\sigma$ above the average
mass-size relation, represent the tail of the satellite size distribution. We
find that each MW analog hosts $N_{\rm UPG} = 0.31\pm 0.05$ ultra-puffy
galaxies on average, which is consistent with but slightly lower than the
observed abundance at this halo mass in the Local Volume. We also construct a
sample of ultra-diffuse galaxies (UDGs) in MW analogs and find an abundance of
$N_{\rm UDG} = 0.44\pm0.05$ per host. With literature results, we confirm that
the UDG abundance scales with the host halo mass following a sublinear power
law. We argue that our definition for ultra-puffy galaxies, which is based on
the mass-size relation, is more physically-motivated than the common definition
of ultra-diffuse galaxies, which depends on surface brightness and size cuts
and thus yields different surface mass density cuts for quenched and
star-forming galaxies.
