Atmospheric errors in interferometric synthetic aperture radar (InSAR)-derived estimates of surface deformation often obscure real displacement signals, especially in mountainous regions. As climate change disproportionately impacts the mountain cryosphere, developing a technique for atmospheric correction that performs well in high-relief terrain is increasingly important. Here, we developed and implemented a statistical machine learning-based atmospheric correction that relies on the differing spatial and topographic characteristics of periglacial features and atmospheric noise. Our correction is applied at the native spatial and temporal resolution of the InSAR data (40 m, 12 days), does not require external atmospheric data, and can correct both stratified and turbulent atmospheric noise. Using Sentinel-1 data from 2015-2022, we trained a convolutional neural network (CNN) on atmospheric noise from 136 short-baseline interferograms and displacement signals from time-series inversion of 337 interferograms. The CNN correction was then tested on a densely connected network of 202 Sentinel-1 interferograms which were inverted to create a displacement time series. We used the Rocky Mountains in Colorado as our training, validation, and testing areas. When applied to our validation data, our correction offers a 690% improvement in performance over a global meteorological reanalysis-based correction and a 209% improvement over a high-pass filter correction. We found that our correction reveals previously hidden time-dependent kinematic behavior of three representative rock glaciers in our testing dataset. Our flexible, robust approach can be used to correct arbitrary InSAR data to analyze subtle surface deformation signals for a range of science and engineering applications.