Castelnuovo-Mumford Regularity in Biprojective Spaces

We define the concept of regularity for bigraded modules and bigraded polynomial ring. In this setting we prove analogs of some of the classical results on $m$-regularity for graded modules over polynomial algebras.Comment: 17 Pages, 2 figure

Curvilinear Base Points, Local Complete Intersection and Kozsul Syzygies in Biprojective Spaces

We prove analogs of results of Cox/Schenck on the structure of certain ideals in the bigraded polynomial ring k[s,u;t,v].Comment: 12 page

Model Selection for High Dimensional Quadratic Regression via Regularization

Quadratic regression (QR) models naturally extend linear models by considering interaction effects between the covariates. To conduct model selection in QR, it is important to maintain the hierarchical model structure between main effects and interaction effects. Existing regularization methods generally achieve this goal by solving complex optimization problems, which usually demands high computational cost and hence are not feasible for high dimensional data. This paper focuses on scalable regularization methods for model selection in high dimensional QR. We first consider two-stage regularization methods and establish theoretical properties of the two-stage LASSO. Then, a new regularization method, called Regularization Algorithm under Marginality Principle (RAMP), is proposed to compute a hierarchy-preserving regularization solution path efficiently. Both methods are further extended to solve generalized QR models. Numerical results are also shown to demonstrate performance of the methods.Comment: 37 pages, 1 figure with supplementary materia