Minimax Optimal Level-Set Estimation

This paper describes a new methodology and associated theoretical analysis for rapid and accurate extraction of level sets of a multivariate function from noisy data. The identification of the boundaries of such sets is an important theoretical problem with applications for digital elevation maps, medical imaging, and pattern recognition. This problem is significantly different from classical segmentation because level-set boundaries may not correspond to singularities or edges in the underlying function; as a result, segmentation methods which rely upon detecting boundaries would be potentially ineffective in this regime. This issue is addressed in this paper through a novel error metric sensitive to both the error in the location of the level-set estimate and the deviation of the function from the critical level. Hoeffding´s inequality is used to derive a novel regularization term that is distinctly different from regularization methods used in conventional image denoising settings. Building upon this foundation, it is possible to derive error performance bounds for the proposed estimator and demonstrate that it exhibits near minimax optimal error decay rates for large classes of level-set problems. The proposed method automatically adapts to the spatially varying regularity of both the boundary of the level set and the underlying function.