  We study the fairness of dimensionality reduction methods for
recommendations. We focus on the established method of principal component
analysis (PCA), which identifies latent components and produces a low-rank
approximation via the leading components while discarding the trailing
components. Prior works have defined notions of "fair PCA"; however, these
definitions do not answer the following question: what makes PCA unfair? We
identify two underlying mechanisms of PCA that induce unfairness at the item
level. The first negatively impacts less popular items, due to the fact that
less popular items rely on trailing latent components to recover their values.
The second negatively impacts the highly popular items, since the leading PCA
components specialize in individual popular items instead of capturing
similarities between items. To address these issues, we develop a
polynomial-time algorithm, Item-Weighted PCA, a modification of PCA that uses
item-specific weights in the objective. On a stylized class of matrices, we
prove that Item-Weighted PCA using a specific set of weights minimizes a
popularity-normalized error metric. Our evaluations on real-world datasets show
that Item-Weighted PCA not only improves overall recommendation quality by up
to $0.1$ item-level AUC-ROC but also improves on both popular and less popular
items.
