The radial velocity technique is one of the two main approaches for detecting planets outside our solar system, often referred to as exoplanets. When a planet orbits a star its gravitational force causes the star to move and this induces a Doppler shift (i.e. the star light appears redder or bluer than expected), and it is this effect that the radial velocity method attempts to detect. Unfortunately, these Doppler signals are typically contaminated by various stellar activity phenomena, such as dark spots on the star surface. We propose a Gaussian process modeling framework to capture this stellar activity and thereby improve detection power for low-mass planets (e.g., Earth-like planets). Our approach builds on previous work in two ways: (i) we use dimension reduction techniques to construct data-driven stellar activity proxies, as opposed to using traditional activity proxies; (ii) we extend the multivariate Gaussian process model of Rajpaul et al. (2015) to a class of models and use a large-scale model selection procedure to find the best model for the particular proxies at hand. Our method results in substantially improved power for planet detection.