  Principal Component Analysis (PCA) is a fundamental tool for data
visualization, denoising, and dimensionality reduction. It is widely popular in
Statistics, Machine Learning, Computer Vision, and related fields. However, PCA
is well known to fall prey to the presence of outliers and often fails to
detect the true underlying low-dimensional structure within the dataset. Recent
supervised learning methods, following the Median of Means (MoM) philosophy,
have shown great success in dealing with outlying observations without much
compromise to their large sample theoretical properties. In this paper, we
propose a PCA procedure based on the MoM principle. Called the Median of Means
Principal Component Analysis (MoMPCA), the proposed method is not only
computationally appealing but also achieves optimal convergence rates under
minimal assumptions. In particular, we explore the non-asymptotic error bounds
of the obtained solution via the aid of Vapnik-Chervonenkis theory and
Rademacher complexity, while granting absolutely no assumption on the outlying
observations. The efficacy of the proposal is also thoroughly showcased through
simulations and real data applications.
