The answer linked below discusses an outlier-tolerant PCA method. Is there a publicly available implementation?
https://stats.stackexchange.com/a/71928/86176
Here's the paper:
Xu, H., Caramanis, C., & Mannor, S. (2013). Outlier-robust PCA: the high-dimensional case. IEEE transactions on information theory, 59(1), 546-572. http://users.ece.utexas.edu/~cmcaram/pubs/HRPCA_Journal.final.pdf
Other reading
Readers may be interested the following discussions of multivariate outlier removal, sparse PCA, and robust covariance estimation, though they do not answer this specific question. (Sparse PCA with corrupted entries is different from outlier-robust PCA with corrupted columns, and OGK does not provide an actual list of outliers.)
Algorithm and R implementation of sparse PCA
What is the best way to identify outliers in multivariate data?
Robust PCA vs. robust Mahalanobis distance for outlier detection
