A structured variable selection problem is considered in which the covariates, divided into predefined groups, activate according to sparse patterns with few nonzero entries per group. Capitalizing on the concept of atomic norm, a composite norm can be properly designed to promote such exclusive group sparsity patterns. The resulting norm lends itself to efficient and flexible regularized optimization algorithms for support recovery, like the proximal algorithm. Moreover, an active set algorithm is proposed that builds the solution by successively including structure atoms into the estimated support. It is also shown that such an algorithm can be tailored to match more rigid structures than plain exclusive group sparsity. Asymptotic consistency analysis (with both the number of parameters as well as the number of groups growing with the observation size) establishes the effectiveness of the proposed solution in terms of signed support recovery under conventional assumptions. Finally, a set of numerical simulations further corroborates the results.

翻译：考虑一个结构化变量选择问题，其中协变量被划分为预定义组，并按照每组中非零条目稀疏的模式激活。基于原子范数的概念，可以恰当设计一种复合范数以促进这种排他性组稀疏模式。所得范数适用于高效且灵活的正则化优化算法（如近端算法）以支持恢复。此外，提出了一种主动集算法，通过逐步将结构原子纳入估计支撑集来构建解。本文还表明，该算法可被定制以匹配比单纯排他性组稀疏更刚性的结构。渐近一致性分析（参数数量与组数均随观测值规模增长）在常规假设下验证了所提解在符号支持恢复方面的有效性。最后，一系列数值模拟进一步佐证了上述结果。