# On the Rotational Invariant $L_1$-Norm PCA

## TL;DR

This paper reinterprets the rotational invariant $L_1$-norm PCA as a gradient-based optimization on Grassmannian manifolds, proving convergence to critical points using the Kurdyka-Łojasiewicz property.

## Contribution

It provides a novel interpretation of robust $L_1$-norm PCA as a gradient descent on Grassmannian manifolds and establishes convergence results for the entire iterative process.

## Key findings

- Reinterpreted $L_1$-norm PCA as a gradient descent on Grassmannian manifolds.
- Proved convergence of all iterates to a critical point using Kurdyka-Łojasiewicz property.
- Unified robust PCA methods under a geometric optimization framework.

## Abstract

Principal component analysis (PCA) is a powerful tool for dimensionality reduction. Unfortunately, it is sensitive to outliers, so that various robust PCA variants were proposed in the literature. Among them the so-called rotational invariant $L_1$-norm PCA is rather popular. In this paper, we reinterpret this robust method as conditional gradient algorithm and show moreover that it coincides with a gradient descent algorithm on Grassmannian manifolds. Based on this point of view, we prove for the first time convergence of the whole series of iterates to a critical point using the Kurdyka-{\L}ojasiewicz property of the energy functional.