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Prior knowledge on properties of a target model often come as discrete orcombinatorial descriptions. This work provides a unified computationalframework for defining norms that promote such structures. More specifically,we develop associated tools for optimization involving such norms given onlythe orthogonal projection oracle onto the non-convex set of desired models. Asan example, we study a norm, which we term the doubly-sparse norm, forpromoting vectors with few nonzero entries taking only a few distinct values.We further discuss how the K-means algorithm can serve as the underlyingprojection oracle in this case and how it can be efficiently represented as aquadratically constrained quadratic program. Our motivation for the study ofthis norm is regularized regression in the presence of rare features whichposes a challenge to various methods within high-dimensional statistics, and inmachine learning in general. The proposed estimation procedure is designed toperform automatic feature selection and aggregation for which we developstatistical bounds. The bounds are general and offer a statistical frameworkfor norm-based regularization. The bounds rely on novel geometric quantities onwhich we attempt to elaborate as well.

New Computational and Statistical Aspects of Regularized Regression with Application to Rare Feature Selection and Aggregation