8 papers
First Proof
Mohammed Abouzaid, Andrew J. Blumberg, Martin Hairer +8
To assess the ability of current AI systems to correctly answer research-level mathematics questions, we share a set of ten math questions which have arisen naturally in the resear…
Geometry and Optimization of Shallow Polynomial Networks
Yossi Arjevani, Joan Bruna, Joe Kileel +2
We study shallow neural networks with monomial activations and output dimension one. The function space for these models can be identified with a set of symmetric tensors with boun…
Efficient Tensor Decomposition via Moment Matrix Extension
Bobby Shi, Julia Lindberg, Joe Kileel
Motivated by a flurry of recent work on efficient tensor decomposition algorithms, we show that the celebrated moment matrix extension algorithm of Brachat, Comon, Mourrain, and Ts…
Covering Number of Real Algebraic Varieties and Beyond: Improved Bounds and Applications
Yifan Zhang, Joe Kileel
Covering numbers are a powerful tool used in the development of approximation algorithms, randomized dimension reduction methods, smoothed complexity analysis, and others. In this…
Norming Sets for Tensor and Polynomial Sketching
Yifan Zhang, Joe Kileel
This paper develops the sketching (i.e., randomized dimension reduction) theory for real algebraic varieties and images of polynomial maps, including, e.g., the set of low rank ten…
Condition numbers in multiview geometry, instability in relative pose estimation, and RANSAC
Hongyi Fan, Joe Kileel, Benjamin Kimia
In this paper, we introduce a general framework for analyzing the numerical conditioning of minimal problems in multiple view geometry, using tools from computational algebra and R…