The non-greedy algorithm for $L_1$-norm PCA proposed in \cite{nie2011robust} is revisited and its convergence properties are studied. The algorithm is first interpreted as a conditional subgradient or an alternating maximization method. By treating it as a conditional subgradient, the iterative points generated by the algorithm will not change in finitely many steps under a certain full-rank assumption; such an assumption can be removed when the projection dimension is one. By treating the algorithm as an alternating maximization, it is proved that the objective value will not change after at most $\left\lceil \frac{F^{\max}}{\tau_0} \right\rceil$ steps. The stopping point satisfies certain optimality conditions. Then, a variant algorithm with improved convergence properties is studied. The iterative points generated by the algorithm will not change after at most $\left\lceil \frac{2F^{\max}}{\tau} \right\rceil$ steps and the stopping point also satisfies certain optimality conditions given a small enough $\tau$. Similar finite-step convergence is also established for a slight modification of the PAMe proposed in \cite{wang2021linear} very recently under a full-rank assumption. Such an assumption can also be removed when the projection dimension is one.

翻译：本文重新审视了文献\cite{nie2011robust}中提出的$L_1$-范数主成分分析的非贪婪算法，并研究了其收敛性质。该算法首先被解释为条件次梯度法或交替最大化方法。若将其视为条件次梯度法，则在满秩假设下，算法生成的迭代点会在有限步内不再变化；当投影维数为1时，该假设可被移除。若将其视为交替最大化，则证明目标值在至多$\left\lceil \frac{F^{\max}}{\tau_0} \right\rceil$步后不再变化，且停止点满足特定最优性条件。随后，本文研究了一种具有改进收敛性质的变体算法，其迭代点在至多$\left\lceil \frac{2F^{\max}}{\tau} \right\rceil$步后不再变化，且在$\tau$足够小时，停止点同样满足特定最优性条件。此外，针对文献\cite{wang2021linear}近期提出的PAMe方法的一个轻微变体，本文在满秩假设下建立了类似的有限步收敛性，且当投影维数为1时，该假设亦可被移除。