Discontinuities in regression functions can reveal important insights. In many contexts, like geographic settings, such discontinuities are multivariate and unknown a priori. We propose a non-parametric regression method that estimates the location and size of discontinuities by segmenting the regression surface. This estimator is based on a convex relaxation of the Mumford-Shah functional, for which we establish identification and convergence. We use it to show that an internet shutdown in India resulted in a reduction of economic activity by 25--35%, greatly surpassing previous estimates and shedding new light on the true cost of such shutdowns for digital economies globally.

翻译：回归函数中的间断性可揭示重要洞察。在诸多背景（如地理情境）下，此类间断点具有多元性且先验未知。我们提出一种非参数回归方法，通过分割回归曲面来估计间断点的位置与幅度。该估计量基于Mumford-Shah泛函的凸松弛，我们为其建立了识别性与收敛性。运用该方法，我们发现印度一次网络断连导致经济活动减少25%至35%，远超此前估计值，为全球数字经济中此类断连的真实成本提供了新视角。