论文标题
凸和非凸LP球投影问题的统一分析
A unified analysis of convex and non-convex lp-ball projection problems
论文作者
论文摘要
将投射到$ \ ell_p $ norm Balls上的任务在统计和机器学习中无处不在，但是可行的算法可用性在很大程度上仅限于$ p = \ weft \ weft \ {0，1,2，\ infty \ infty \ right \ right \} $的特殊情况。在本文中，我们介绍了用于投影到一般$ p> 0 $的$ \ ell_p $球上的小说可扩展方法。对于$ p \ geq1 $，我们通过双牛顿方法解决了单变量的拉格朗日二重奏。然后，我们仔细设计了一种$ p <1 $的二分线方法，在非凸案例中提供了零或小对偶差的理论和经验证据。我们的贡献的成功得到了经验的彻底评估，并应用于大规模的正规化多任务学习和压缩感测。
The task of projecting onto $\ell_p$ norm balls is ubiquitous in statistics and machine learning, yet the availability of actionable algorithms for doing so is largely limited to the special cases of $p = \left\{ 0, 1,2, \infty \right\}$. In this paper, we introduce novel, scalable methods for projecting onto the $\ell_p$ ball for general $p>0$. For $p \geq1 $, we solve the univariate Lagrangian dual via a dual Newton method. We then carefully design a bisection approach for $p<1$, presenting theoretical and empirical evidence of zero or a small duality gap in the non-convex case. The success of our contributions is thoroughly assessed empirically, and applied to large-scale regularized multi-task learning and compressed sensing.
