  Recently, linear regression models, such as EASE and SLIM, have shown to
often produce rather competitive results against more sophisticated deep
learning models. On the other side, the (weighted) matrix factorization
approaches have been popular choices for recommendation in the past and widely
adopted in the industry. In this work, we aim to theoretically understand the
relationship between these two approaches, which are the cornerstones of
model-based recommendations. Through the derivation and analysis of the
closed-form solutions for two basic regression and matrix factorization
approaches, we found these two approaches are indeed inherently related but
also diverge in how they "scale-down" the singular values of the original
user-item interaction matrix. This analysis also helps resolve the questions
related to the regularization parameter range and model complexities. We
further introduce a new learning algorithm in searching (hyper)parameters for
the closed-form solution and utilize it to discover the nearby models of the
existing solutions. The experimental results demonstrate that the basic models
and their closed-form solutions are indeed quite competitive against the
state-of-the-art models, thus, confirming the validity of studying the basic
models. The effectiveness of exploring the nearby models are also
experimentally validated.
