  Linear autoencoder models learn an item-to-item weight matrix via convex
optimization with L2 regularization and zero-diagonal constraints. Despite
their simplicity, they have shown remarkable performance compared to
sophisticated non-linear models. This paper aims to theoretically understand
the properties of two terms in linear autoencoders. Through the lens of
singular value decomposition (SVD) and principal component analysis (PCA), it
is revealed that L2 regularization enhances the impact of high-ranked PCs.
Meanwhile, zero-diagonal constraints reduce the impact of low-ranked PCs,
leading to performance degradation for unpopular items. Inspired by this
analysis, we propose simple-yet-effective linear autoencoder models using
diagonal inequality constraints, called Relaxed Linear AutoEncoder (RLAE) and
Relaxed Denoising Linear AutoEncoder (RDLAE). We prove that they generalize
linear autoencoders by adjusting the degree of diagonal constraints.
Experimental results demonstrate that our models are comparable or superior to
state-of-the-art linear and non-linear models on six benchmark datasets; they
significantly improve the accuracy of long-tail items. These results also
support our theoretical insights on regularization and diagonal constraints in
linear autoencoders.
