  Sparse principal component analysis (PCA) aims at mapping large dimensional
data to a linear subspace of lower dimension. By imposing loading vectors to be
sparse, it performs the double duty of dimension reduction and variable
selection. Sparse PCA algorithms are usually expressed as a trade-off between
explained variance and sparsity of the loading vectors (i.e., number of
selected variables). As a high explained variance is not necessarily synonymous
with relevant information, these methods are prone to select irrelevant
variables. To overcome this issue, we propose an alternative formulation of
sparse PCA driven by the false discovery rate (FDR). We then leverage the
Terminating-Random Experiments (T-Rex) selector to automatically determine an
FDR-controlled support of the loading vectors. A major advantage of the
resulting T-Rex PCA is that no sparsity parameter tuning is required. Numerical
experiments and a stock market data example demonstrate a significant
performance improvement.
