Robust PCA via Regularized Reaper with a Matrix-Free Proximal Algorithm.

Area of Science:

• Data Science
• Machine Learning
• Optimization

Background:

• Principal Component Analysis (PCA) is sensitive to outliers, necessitating robust variants.
• Existing robust PCA methods like reaper face limitations with high-dimensional data due to computational constraints.

Purpose of the Study:

• To develop a regularized version of the reaper model for robust PCA.
• To enable robust PCA on high-dimensional datasets, common in fields like image processing.

Main Methods:

• A regularized reaper model is proposed, penalizing the nuclear norm to enforce sparsity of principal components.
• A matrix-free primal-dual algorithm coupled with a Lanczos process is developed for efficient minimization.
• Integration with the L-curve method for subspace reconstruction when the number of components is bounded.

Main Results:

• The proposed method effectively handles high-dimensional data, overcoming limitations of previous approaches.
• The matrix-free algorithm demonstrates efficiency and suitability for large-scale robust PCA.
• Numerical examples validate the performance of the developed algorithm.

Conclusions:

• This work presents the first efficient convex variational method for robust PCA applicable to high-dimensional data.
• The regularized reaper model and its associated algorithm offer a significant advancement in robust dimensionality reduction.