  Learning user preferences for products based on their past purchases or
reviews is at the cornerstone of modern recommendation engines. One
complication in this learning task is that some users are more likely to
purchase products or review them, and some products are more likely to be
purchased or reviewed by the users. This non-uniform pattern degrades the power
of many existing recommendation algorithms, as they assume that the observed
data are sampled uniformly at random among user-product pairs. In addition,
existing literature on modeling non-uniformity either assume user interests are
independent of the products, or lack theoretical understanding. In this paper,
we first model the user-product preferences as a partially observed matrix with
non-uniform observation pattern. Next, building on the literature about
low-rank matrix estimation, we introduce a new weighted trace-norm penalized
regression to predict unobserved values of the matrix. We then prove an upper
bound for the prediction error of our proposed approach. Our upper bound is a
function of a number of parameters that are based on a certain weight matrix
that depends on the joint distribution of users and products. Utilizing this
observation, we introduce a new optimization problem to select a weight matrix
that minimizes the upper bound on the prediction error. The final product is a
new estimator, NU-Recommend, that outperforms existing methods in both
synthetic and real datasets. Our approach aims at accurate predictions for all
users while prioritizing fairness. To achieve this, we employ a bias-variance
tradeoff mechanism that ensures good overall prediction performance without
compromising the predictive accuracy for less active users.

Information Retrieval;Machine Learning