Algebraic methods in sum-product phenomena

Sum-product Theorems in Algebraic Number Fields

SUM-PRODUCT PHENOMENA: p-ADIC CASE

The sum-product phenomena over a finite extension K of Qp is explored. The main feature of the results is the fact that the implied constants are independent p.

SUM - PRODUCT PHENOMENA IN F p : A BRIEF INTRODUCTION

These notes arose from my Cambridge Part III course on Additive Combinatorics, given in Lent Term 2009. The aim was to understand the simplest proof of the Bourgain-Glibichuk-Konyagin bounds for exponential sums over subgroups. As a byproduct one obtains a clean proof of the Bourgain-Katz-Tao theorem on the sumproduct phenomenon in Fp. The arguments are essentially extracted from Bourgain’s pap...

Sum-Product-Quotient Networks

We present a novel tractable generative model that extends Sum-Product Networks (SPNs) and significantly boosts their power. We call it Sum-Product-Quotient Networks (SPQNs), whose core concept is to incorporate conditional distributions into the model by direct computation using quotient nodes, e.g. P (A|B)= (A,B) P (B) . We provide sufficient conditions for the tractability of SPQNs that gene...

Credal Sum-Product Networks

Sum-product networks are a relatively new and increasingly popular class of (precise) probabilistic graphical models that allow for marginal inference with polynomial effort. As with other probabilistic models, sum-product networks are often learned from data and used to perform classification. Hence, their results are prone to be unreliable and overconfident. In this work, we develop credal su...