Adelic amoebas disjoint from open halfspaces

Adelic Amoebas Disjoint from Open Halfspaces

We show that a conjecture of Einsiedler, Kapranov, and Lind on adelic amoebas of subvarieties of tori and their intersections with open halfspaces of complementary dimension is false for subvarieties of codimension greater than one that have degenerate projections to smaller dimensional tori. We prove a suitably modified version of the conjecture using algebraic methods, functoriality of tropic...

Generalizing Halfspaces Generalizing Halfspaces

Restricted-orientation convexity is the study of geometric objects whose intersection with lines from some fixed set is empty or connected. We have studied the properties of restricted-orientation convex sets and demonstrated that this notion is a generalization of standard convexity. We now describe a restricted-orientation generalization of halfspaces and explore properties of these generaliz...

Open Problem: The Statistical Query Complexity of Learning Sparse Halfspaces

We consider the long-open problem of attribute-efficient learning of halfspaces. In this problem the learner is given random examples labeled by an unknown halfspace function f on R. Further f is r-sparse, that is it depends on at most r out of n variables. An attribute-efficient learning algorithm is an algorithm that can output a hypothesis close to f using a polynomial in r and log n number ...

Computing Amoebas

Generalizing Halfspaces