We introduce a strict saddle property for $\ell_p$ regularized functions, and propose an iterative reweighted $\ell_1$ algorithm to solve the $\ell_p$ regularized problems. The algorithm is guaranteed to converge only to local minimizers when randomly initialized. The strict saddle property is shown generic on these sparse optimization problems. Those analyses as well as the proposed algorithm can be easily extended to general nonconvex regularized problems.

翻译：我们引入了一个针对$\ell_p$正则化函数的严格鞍点性质，并提出了一种迭代重加权$\ell_1$算法来求解$\ell_p$正则化问题。该算法在随机初始化时保证仅收敛至局部极小值点。我们证明了严格鞍点性质在这些稀疏优化问题上具有普适性。上述分析及所提算法可轻松推广至一般的非凸正则化问题。