Maximum-Variance Total Variation Denoising for Interpretable Spatial Smoothing

We consider the problem of spatial regression where interpretability of the model is a high priority. Such problems appear frequently in a diverse set of fields from climatology to epidemiology to predictive policing. For cognitive, logistical, and organizational reasons, humans tend to infer regions or neighborhoods of constant value, often with sharp discontinuities between regions, and then assign resources on a per-region basis. Automating this smoothing process presents a unique challenge for spatial smoothing algorithms, which tend to assume stationarity and smoothness everywhere. To address this problem, we propose Maximum Variance Total Variation (MVTV) denoising, a novel method for interpretable nonlinear spatial regression. MVTV divides the feature space into blocks of constant value and smooths the value of all blocks jointly via a convex optimization routine. Our method is fully data-adaptive and incorporates highly robust routines for tuning all hyperparameters automatically. We compare our approach against the existing CART and CRISP methods via both a complexity-accuracy tradeoff metric and a human study, demonstrating that that MVTV is a more powerful and interpretable method.