10 papers · 1 filter
Critical Points of Degenerate Metrics on Algebraic Varieties: A Tale of Overparametrization
Giovanni Luca Marchetti, Erin Connelly, Paul Breiding +1
We study the critical points over an algebraic variety of an optimization problem defined by a quadratic objective that is degenerate. This scenario arises in machine learning when…
Homotopy Iterators
Paul Breiding, Taylor Brysiewicz, Hannah Friedman
We introduce the concept of homotopy iterators for performing polynomial homotopy continuation tasks in a memory efficient manner. The main idea is to push forward an iterator for…
Critical Curvature of Algebraic Surfaces in Three-Space
Paul Breiding, Kristian Ranestad, Madeleine Weinstein
We study the curvature of a smooth algebraic surface of degree from the point of view of algebraic geometry. More precisely, we consider umbilical points…
A short proof for the parameter continuation theorem
Viktoriia Borovik, Paul Breiding
The Parameter Continuation Theorem is the theoretical foundation for polynomial homotopy continuation, which is one of the main tools in computational algebraic geometry. In this n…
Typical ranks of random order-three tensors
Paul Breiding, Sarah Eggleston, Andrea Rosana
In this paper we study typical ranks of real tensors. In the case the typical ranks are contained in ,…
Certifying zeros of polynomial systems using interval arithmetic
Paul Breiding, Kemal Rose, Sascha Timme
We establish interval arithmetic as a practical tool for certification in numerical algebraic geometry. Our software HomotopyContinuation.jl now has a built-in function certify, wh…