Exact Rotation Invariant Robust PCA

In this work, we address the challenge of estimating principal components in the presence of outliers, a problem commonly referred to as robust principal component analysis (PCA). Traditional PCA minimizes the Euclidean norm, making it vulnerable to outliers. To enhance robustness, earlier approaches replaced the Euclidean norm with the l<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf>-norm, which improved resilience to outliers but did not fully eliminate their influence. Subsequent studies proposed the use of the l<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf>-norm (0 < p < 1), which further reduced the impact of outliers. In this paper, we advance this line of research by replacing the l<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf>-norm with the l<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf>-norm, effectively eliminating the influence of outliers in the estimation process. To maintain rotational invariance, we introduce the l<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2,0</inf>-norm, applying the l<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf>-norm and the l<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf>-norm selectively across features and samples. The proposed formulations are solved using augmented Lagrangians followed by alternating minimization. Extensive experiments on datasets from the UCI Machine Learning Repository demonstrate that our method outperforms previous robust PCA formulations, including R<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf>-PCA, principal component pursuit, and the recently introduced median-of-means-based robust PCA approach.