标题： 快速投影到混合范数球及其应用 显示英文标题

标题： Fast projections onto mixed-norm balls with applications

摘要： 联合稀疏性为特征选择提供了强大的结构线索，特别是在那些预期表现出“分组”行为的变量方面。 这种行为通常通过组套索（group-lasso）、多任务套索（multitask lasso）及相关方法来建模，在这些方法中，特征选择是通过混合范数实现的。 一些基于混合范数的稀疏模型受到了广泛关注，并且在某些情况下还存在高效的算法。 令人惊讶的是，一些受限的稀疏模型似乎缺乏可扩展的算法。 我们通过提出批量优化和在线（随机梯度）优化方法来解决这一不足，这两种方法都依赖于对混合范数球体的有效投影。 我们通过将这些方法应用于多任务套索来展示它们。 最后，我们提到一些开放性问题。 显示英文摘要

摘要： Joint sparsity offers powerful structural cues for feature selection, especially for variables that are expected to demonstrate a "grouped" behavior. Such behavior is commonly modeled via group-lasso, multitask lasso, and related methods where feature selection is effected via mixed-norms. Several mixed-norm based sparse models have received substantial attention, and for some cases efficient algorithms are also available. Surprisingly, several constrained sparse models seem to be lacking scalable algorithms. We address this deficiency by presenting batch and online (stochastic-gradient) optimization methods, both of which rely on efficient projections onto mixed-norm balls. We illustrate our methods by applying them to the multitask lasso. We conclude by mentioning some open problems.