  Doppler spectroscopy is a powerful tool for discovering and characterizing
exoplanets. For decades, the standard approach to extracting radial velocities
(RVs) has been to cross-correlate observed spectra with a weighted template
mask. While still widely used, this approach is known to suffer numerous
drawbacks, and so in recent years increasing attention has been paid to
developing new and improved ways of extracting RVs. In this proof-of-concept
paper we present a simple yet powerful approach to RV extraction. We use
Gaussian processes to model and align all pairs of spectra with each other;
without constructing a template, we combine pairwise RV shifts to produce
accurate differential stellar RVs. Doing this on a highly-localized basis
enables a data-driven approach to identifying and mitigating spectral
contamination, even without the input of any prior astrophysical knowledge. We
show that a crude implementation of this method applied to an inactive standard
star yields RVs with comparable precision to and significantly lower rms
variation than RVs from industry-standard pipelines. Though amenable to
numerous improvements, even in its basic form presented here our method could
facilitate the study of smaller planets around a wider variety of stars than
has previously been possible.
