The Nash Problem from Geometric and Topological Perspective

We survey the proof of the Nash conjecture for surfaces and show how geometric and topological ideas developed in previous articles by the authors influenced it. Later, we summarize the main ideas in the higher dimensional statement and proof by de Fernex and Docampo. We end the paper by explaining later developments on generalized Nash problem and on Kollar and Nemethi holomorphic arcs

Robust priors for regularized regression

Induction benefits from useful priors. Penalized regression approaches, like ridge regression, shrink weights toward zero but zero association is usually not a sensible prior. Inspired by simple and robust decision heuristics humans use, we constructed non-zero priors for penalized regression models that provide robust and interpretable solutions across several tasks. Our approach enables estimates from a constrained model to serve as a prior for a more general model, yielding a principled way to interpolate between models of differing complexity. We successfully applied this approach to a number of decision and classification problems, as well as analyzing simulated brain imaging data. Models with robust priors had excellent worst-case performance. Solutions followed from the form of the heuristic that was used to derive the prior. These new algorithms can serve applications in data analysis and machine learning, as well as help in understanding how people transition from novice to expert performance

The Nash Problem from a Geometric and Topological Perspective

We survey the proof of the Nash conjecture for surfaces and show how geometric and topological ideas developed in previous articles by the au- thors influenced it. Later we summarize the main ideas in the higher dimen- sional statement and proof by de Fernex and Docampo. We end the paper by explaining later developments on generalized Nash problem and on Koll ́ar and Nemethi holomorphic arcs

Representation of surface homeomorphisms by tête-à-tête graphs

We use tête-à-tête graphs as defined by N. A'campo and extended versions to codify all periodic mapping classes of an orientable surface with non-empty boundary, improving work of N. A'Campo and C. Graf. We also introduce the notion of mixed tête-à-tête graphs to model some pseudo-periodic homeomorphisms. In particular we are able to codify the monodromy of any irreducible plane curve singularity. The work ends with an appendix that studies all the possible combinatorial structures that make a given filtered metric ribbon graph with some regularity conditions into a mixed tête-à-tête graph

Coupling multiple views of relations for recommendation

© Springer International Publishing Switzerland 2015. Learning user/item relation is a key issue in recommender system, and existing methods mostly measure the user/item relation from one particular aspect, e.g., historical ratings, etc. However, the relations between users/items could be influenced by multifaceted factors, so any single type of measure could get only a partial view of them. Thus it is more advisable to integrate measures from different aspects to estimate the underlying user/item relation. Furthermore, the estimation of underlying user/item relation should be optimal for current task. To this end, we propose a novel model to couple multiple relations measured on different aspects, and determine the optimal user/item relations via learning the optimal way of integrating these relation measures. Specifically, matrix factorization model is extended in this paper by considering the relations between latent factors of different users/items. Experiments are conducted and our method shows good performance and outperforms other baseline methods