  Unsupervised representation learning is one of the most important problems in
machine learning. Recent promising methods are based on contrastive learning.
However, contrastive learning often relies on heuristic ideas, and therefore it
is not easy to understand what contrastive learning is doing. This paper
emphasizes that density ratio estimation is a promising goal for unsupervised
representation learning, and promotes understanding to contrastive learning.
Our primal contribution is to theoretically show that density ratio estimation
unifies three frameworks for unsupervised representation learning: Maximization
of mutual information (MI), nonlinear independent component analysis (ICA) and
a novel framework for estimation of a lower-dimensional nonlinear subspace
proposed in this paper. This unified view clarifies under what conditions
contrastive learning can be regarded as maximizing MI, performing nonlinear ICA
or estimating the lower-dimensional nonlinear subspace in the proposed
framework. Furthermore, we also make theoretical contributions in each of the
three frameworks: We show that MI can be maximized through density ratio
estimation under certain conditions, while our analysis for nonlinear ICA
reveals a novel insight for recovery of the latent source components, which is
clearly supported by numerical experiments. In addition, some theoretical
conditions are also established to estimate a nonlinear subspace in the
proposed framework. Based on the unified view, we propose two practical methods
for unsupervised representation learning through density ratio estimation: The
first method is an outlier-robust method for representation learning, while the
second one is a sample-efficient nonlinear ICA method. Finally, we numerically
demonstrate usefulness of the proposed methods in nonlinear ICA and through
application to a downstream task for classification.
