34 citations · 65 across the 16 of their papers we have counts for
18 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…
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…
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…
The Fascinating World of 2 2 2 Tensors: Its Geometry and Optimization Challenges
Gabriel H. Brown, Joe Kileel, Tamara G. Kolda
This educational article highlights the geometric and algebraic complexities that distinguish tensors from matrices, to supplement coverage in advanced courses on linear algebra, m…
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…
Tensor-Based Synchronization and the Low-Rankness of the Block Trifocal Tensor
Daniel Miao, Gilad Lerman, Joe Kileel
The block tensor of trifocal tensors provides crucial geometric information on the three-view geometry of a scene. The underlying synchronization problem seeks to recover camera po…
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…
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…
Moment Estimation for Nonparametric Mixture Models Through Implicit Tensor Decomposition
Yifan Zhang, Joe Kileel
We present an alternating least squares type numerical optimization scheme to estimate conditionally-independent mixture models in , without parameterizing the distri…
Snapshot of Algebraic Vision
Joe Kileel, Kathlén Kohn
In this survey article, we present interactions between algebraic geometry and computer vision, which have recently come under the header of algebraic vision. The subject has given…
Scalable symmetric Tucker tensor decomposition
Ruhui Jin, Joe Kileel, Tamara G. Kolda +1
We study the best low-rank Tucker decomposition of symmetric tensors. The motivating application is decomposing higher-order multivariate moments. Moment tensors have special struc…
Tensor Moments of Gaussian Mixture Models: Theory and Applications
João M. Pereira, Joe Kileel, Tamara G. Kolda
Gaussian mixture models (GMMs) are fundamental tools in statistical and data sciences. We study the moments of multivariate Gaussians and GMMs. The -th moment of an -dimensio…
On the Instability of Relative Pose Estimation and RANSAC's Role
Hongyi Fan, Joe Kileel, Benjamin Kimia
In this paper we study the numerical instabilities of the 5- and 7-point problems for essential and fundamental matrix estimation in multiview geometry. In both cases we characteri…
Landscape analysis of an improved power method for tensor decomposition
Joe Kileel, Timo Klock, João M. Pereira
In this work, we consider the optimization formulation for symmetric tensor decomposition recently introduced in the Subspace Power Method (SPM) of Kileel and Pereira. Unlike popul…
Symmetry Breaking in Symmetric Tensor Decomposition
Yossi Arjevani, Joan Bruna, Michael Field +3
In this note, we consider the highly nonconvex optimization problem associated with computing the rank decomposition of symmetric tensors. We formulate the invariance properties of…
Subspace power method for symmetric tensor decomposition
Joe Kileel, João M. Pereira
We introduce the Subspace Power Method (SPM) for calculating the CP decomposition of low-rank real symmetric tensors. This algorithm calculates one new CP component at a time, alte…
On the Expressive Power of Deep Polynomial Neural Networks
Joe Kileel, Matthew Trager, Joan Bruna
We study deep neural networks with polynomial activations, particularly their expressive power. For a fixed architecture and activation degree, a polynomial neural network defines…
Numerical Implicitization
Justin Chen, Joe Kileel
We present the package for , which allows for user-friendly computation of the invariants of the image of a polynomial map,…