  Matrix factorization (MF) plays an important role in a wide range of machine
learning and data mining models. MF is commonly used to obtain item embeddings
and feature representations due to its ability to capture correlations and
higher-order statistical dependencies across dimensions. In many applications,
the categories of items exhibit a hierarchical tree structure. For instance,
human diseases can be divided into coarse categories, e.g., bacterial, and
viral. These categories can be further divided into finer categories, e.g.,
viral infections can be respiratory, gastrointestinal, and exanthematous viral
diseases. In e-commerce, products, movies, books, etc., are grouped into
hierarchical categories, e.g., clothing items are divided by gender, then by
type (formal, casual, etc.). While the tree structure and the categories of the
different items may be known in some applications, they have to be learned
together with the embeddings in many others. In this work, we propose eTREE, a
model that incorporates the (usually ignored) tree structure to enhance the
quality of the embeddings. We leverage the special uniqueness properties of
Nonnegative MF (NMF) to prove identifiability of eTREE. The proposed model not
only exploits the tree structure prior, but also learns the hierarchical
clustering in an unsupervised data-driven fashion. We derive an efficient
algorithmic solution and a scalable implementation of eTREE that exploits
parallel computing, computation caching, and warm start strategies. We showcase
the effectiveness of eTREE on real data from various application domains:
healthcare, recommender systems, and education. We also demonstrate the
meaningfulness of the tree obtained from eTREE by means of domain experts
interpretation.
