J.-Ph. Vert 

Titre
Collaborative filtering in Hilbert spaces with spectral regularization

Abstract
Collaborative Filtering (CF) refers to the task of learning preferences of 
customers for products, such as books or movies, from a set of known 
preferences. More formally, this can be seen as the task of filling missing 
entries in a matrix where some entries are known. A standard approach 
to CF is to find a low rank approximation to the matrix. This problem is 
computationally difficult and some authors have proposed recently to search 
instead for a low trace norm matrix, which results in a convex optimization 
problem. We generalize this approach to the estimation of a compact operator, 
of which matrix estimation is a special case. We develop a notion of spectral 
regularization which captures both rank constraint and trace norm 
regularization, as well as many others. The major advantage of this approach 
is that it provides a natural method of utilizing side-information, such as 
age and gender, about the customers (or objects) in question - a formerly 
challenging limitation of the low-rank approach. We provide a number of 
algorithms, and test our results on a standard CF dataset with promising 
results. This is a joint work with Jacob Abernethy (UC Berkeley), 
Francis Bach (INRIA), and Theodoros Evgeniou (INSEAD).