On Finite-Step Convergence of the Non-Greedy Algorithm for $L_1$-Norm PCA and Beyond

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.