  A popular robust alternative of the classic principal component analysis
(PCA) is the $\ell_1$-norm PCA (L1-PCA), which aims to find a subspace that
captures the most variation in a dataset as measured by the $\ell_1$-norm.
L1-PCA has shown great promise in alleviating the effect of outliers in data
analytic applications. However, it gives rise to a challenging non-smooth
non-convex optimization problem, for which existing algorithms are either not
scalable or lack strong theoretical guarantees on their convergence behavior.
In this paper, we propose a proximal alternating minimization method with
extrapolation (PAMe) for solving a two-block reformulation of the L1-PCA
problem. We then show that for both the L1-PCA problem and its two-block
reformulation, the Kurdyka-\L ojasiewicz exponent at any of the limiting
critical points is $1/2$. This allows us to establish the linear convergence of
the sequence of iterates generated by PAMe and to determine the criticality of
the limit of the sequence with respect to both the L1-PCA problem and its
two-block reformulation. To complement our theoretical development, we show via
numerical experiments on both synthetic and real-world datasets that PAMe is
competitive with a host of existing methods. Our results not only significantly
advance the convergence theory of iterative methods for L1-PCA but also
demonstrate the potential of our proposed method in applications.
