Group-sparse Embeddings in Collective Matrix Factorization
Authors: Arto Klami, Guillaume Bouchard, Abhishek Tripathi
ICLR 2014

Abstract: CMF is a technique for simultaneously learning low-rank representations based on a collection of matrices with shared entities. A typical example is the joint modeling of user-item, item-property, and user-feature matrices in a recommender system. The key idea in CMF is that the embeddings are shared across the matrices, which enables transferring information between them. The existing solutions, however, break down when the individual matrices have low-rank structure not shared with others. In this work we present a novel CMF solution that allows each of the matrices to have a separate low-rank structure that is independent of the other matrices, as well as structures that are shared only by a subset of them. We compare MAP and variational Bayesian solutions based on alternating optimization algorithms and show that the model automatically infers the nature of each factor using group-wise sparsity. Our approach supports in a principled way continuous, binary and count observations and is efficient for sparse matrices involving missing data. We illustrate the solution on a number of examples, focusing in particular on an interesting use-case of augmented multi-view learning.


Peer review:
[Review]:
Collective matrix factorization (CMF) is a method for learning entity embeddings by jointly factorizing a collection of matrices, each of which describes a relationship between entities of two types. The set of rows/columns of a matrix corresponds to an entity type, while each row/column of the matrix corresponds to a different entity of the type. This approach assumes that all dimensions of the embeddings of entities of a particular type are relevant for modelling all the relationships they are involved in. This paper extends CMF by relaxing this assumption and allowing embeddings to have dimensions used for modelling only some of the relationships the entities are involved in. This is achieved by extending the model to include a separate precision for each embedding dimension of each type to encourage group-sparse solutions. The authors propose training the resulting models using a variational Bayes algorithm and show how to handle both Gaussian and non-Gaussian observations.

The paper is nicely written and makes a small but novel extension to CMF. The resulting approach is simple, and seems scalable and widely applicable. The experiments are fairly small-scale but are sufficient for illustrating the advantages of the method.

Corrections:
In the section dealing non-Gaussian observations, the pseudo-data is referred to as Y instead of Z_m.

The description of variational Bayes as 'minimizing the KL divergence between the observation probability and ...' is not quite right, as it seems to describe KL(P||Q) instead of KL(Q||P).

Section 8: 'unability' -> 'inability'