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 certain 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 PCA 的非贪婪算法，并研究了其收敛性质。首先将该算法解释为条件次梯度法或交替最大化法。将其视为条件次梯度法时，在一定的满秩假设下，算法生成的迭代点会在有限步内保持不变；当投影维度为一时，该假设可被移除。将其视为交替最大化法时，证明了目标值在至多 $\left\lceil \frac{F^{\max}}{\tau_0} \right\rceil$ 步后不再变化，且停止点满足某些最优性条件。随后，本文研究了一种收敛性质更优的变体算法，其迭代点在至多 $\left\lceil \frac{2F^{\max}}{\tau} \right\rceil$ 步后保持不变，并且当 $\tau$ 足够小时，停止点也满足某些最优性条件。本文还建立了近期文献\cite{wang2021linear}提出的 PAMe 的轻微变体在满秩假设下的类似有限步收敛性质；当投影维度为一时，该假设同样可被移除。