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, 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. Inferences 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 inferences about their surface properties.

多亏了像开普勒和苔丝这样的任务，我们现在可以获得数以万计的高精度、快节奏和长基线的恒星光度观测。原则上，这些光变曲线编码了大量关于恒星变异性的信息，特别是星斑的分布和其表面的其他特征。不幸的是，从旋转光曲线推断恒星表面特性的问题是出了名的不适定的，因为它通常没有唯一的解决方案。因此，关于斑点的数量、大小、对比度和位置的推断可能非常依赖于模型的假设、正则化方案或先验。本文的目标有两个：(1) 探索影响恒星光曲线“反转”问题的各种简并性及其对在未解析的光度测量值下可以和不能从恒星表面学到什么的影响，以及 (2) 激发对光曲线的集合分析许多明星同时作为强大的数据驱动替代品，以替代文献中采用的常见先验。我们进一步推导出零空间对恒星倾角和边缘变暗的依赖性的新结果，并表明单波段光度测量不能在不使用强先验的情况下唯一地限制像总光斑覆盖率这样的数量。这是致力于开发用于分析恒星光变曲线和光谱时间序列的新算法和工具的系列论文中的第一篇，