  Thanks to missions like Kepler and TESS, we now have access to tens of
thousands of high precision, fast cadence, and long baseline stellar
photometric observations. In principle, these light curves encode a vast amount
of information about stellar variability and, in particular, about the
distribution of starspots and other features on their surfaces. Unfortunately,
the problem of inferring stellar surface properties from a rotational light
curve is famously ill-posed, as it often does not admit a unique solution.
Inference about the number, size, contrast, and location of spots can therefore
depend very strongly on the assumptions of the model, the regularization
scheme, or the prior. The goal of this paper is twofold: (1) to explore the
various degeneracies affecting the stellar light curve "inversion" problem and
their effect on what can and cannot be learned from a stellar surface given
unresolved photometric measurements; and (2) to motivate ensemble analyses of
the light curves of many stars at once as a powerful data-driven alternative to
common priors adopted in the literature. We further derive novel results on the
dependence of the null space on stellar inclination and limb darkening and show
that single-band photometric measurements cannot uniquely constrain quantities
like the total spot coverage without the use of strong priors. This is the
first in a series of papers devoted to the development of novel algorithms and
tools for the analysis of stellar light curves and spectral time series, with
the explicit goal of enabling statistically robust inference about their
surface properties.
