In standard Principal Component Analysis (PCA), we center the data by subtracting the mean of each coordinate before computing the covariance matrix and solving for the principal components.
I am wondering: what would happen if, instead of centering the data with the mean, we center it using the coordinate-wise median?
In particular: • Would the general PCA optimization problem (e.g., minimizing reconstruction error) still look the same? • Would the principal components still correspond to directions of maximal variance?
I understand that median centering is more robust to outliers, but I am trying to understand precisely how the mathematical structure and meaning of PCA would change.
Thanks for any clarification!
