A trace formula approach to control theorems for overconvergent automorphic forms

We present an approach to proving control theorems for overconvergent automorphic forms on some Harris-Taylor unitary Shimura varieties based on a comparison between the rigid coho- mology of the multiplicative ordinary locus and the rigid cohomology of the overlying Igusa tower, the latter which may be computed using the Harris-Taylor version of the Langlands-Kottwitz method. We also prove a higher level version, generalizing work of Coleman.Comment: 25 pages. Main results strengthened, higher level version include

A New Approach to Collaborative Filtering: Operator Estimation with Spectral Regularization

We present a general approach for collaborative filtering (CF) using spectral regularization to learn linear operators from "users" to the "objects" they rate. Recent low-rank type matrix completion approaches to CF are shown to be special cases. However, unlike existing regularization based CF methods, our approach can be used to also incorporate information such as attributes of the users or the objects -- a limitation of existing regularization based CF methods. We then provide novel representer theorems that we use to develop new estimation methods. We provide learning algorithms based on low-rank decompositions, and test them on a standard CF dataset. The experiments indicate the advantages of generalizing the existing regularization based CF methods to incorporate related information about users and objects. Finally, we show that certain multi-task learning methods can be also seen as special cases of our proposed approach