no code implementations • 5 Apr 2016 • Ke Ye, Lek-Heng Lim
The goal of cryo-electron microscopy (EM) is to reconstruct the 3-dimensional structure of a molecule from a collection of its 2-dimensional projected images.
1 code implementation • ICML 2018 • Liwen Zhang, Gregory Naitzat, Lek-Heng Lim
Among other things, we deduce that feedforward ReLU neural networks with one hidden layer can be characterized by zonotopes, which serve as building blocks for deeper networks; we relate decision boundaries of such neural networks to tropical hypersurfaces, a major object of study in tropical geometry; and we prove that linear regions of such neural networks correspond to vertices of polytopes associated with tropical rational functions.
1 code implementation • 18 Dec 2018 • Tingran Gao, Lek-Heng Lim, Ke Ye
We introduce in this paper a manifold optimization framework that utilizes semi-Riemannian structures on the underlying smooth manifolds.
Optimization and Control Numerical Analysis 90C30, 53C50, 53B30, 49M05, 49M15 F.2.1; G.1.6
no code implementations • 2 Jul 2019 • Lek-Heng Lim, Mateusz Michalek, Yang Qi
A high-level explanation is like that for the nonexistence of best rank-$r$ approximations of higher-order tensors --- the set of parameters is not a closed set --- but the geometry involved for best $k$-layer neural networks approximations is more subtle.
no code implementations • 13 Apr 2020 • Gregory Naitzat, Andrey Zhitnikov, Lek-Heng Lim
We study how the topology of a data set $M = M_a \cup M_b \subseteq \mathbb{R}^d$, representing two classes $a$ and $b$ in a binary classification problem, changes as it passes through the layers of a well-trained neural network, i. e., with perfect accuracy on training set and near-zero generalization error ($\approx 0. 01\%$).
no code implementations • 2 Jun 2020 • Zehua Lai, Lek-Heng Lim
Our approach relies on the noncommutative Positivstellensatz, which allows us to reduce the conjectured inequality to a semidefinite program and the validity of the conjecture to certain bounds for the optimum values, which we show are false as soon as $n = 5$.
no code implementations • 15 May 2022 • Lek-Heng Lim, Bradley J. Nelson
We explain equivariant neural networks, a notion underlying breakthroughs in machine learning from deep convolutional neural networks for computer vision to AlphaFold 2 for protein structure prediction, without assuming knowledge of equivariance or neural networks.
no code implementations • 25 Nov 2022 • Yucong Liu, Simiao Jiao, Lek-Heng Lim
It is well-known that any matrix $A$ has an LU decomposition.
no code implementations • 28 Nov 2022 • Minda Zhao, Zehua Lai, Lek-Heng Lim
Is it possible for a first-order method, i. e., only first derivatives allowed, to be quadratically convergent?
no code implementations • ICML 2020 • Zehua Lai, Lek-Heng Lim
Stochastic optimization algorithms have become indispensable in machine learning.
no code implementations • ICML 2020 • Zehua Lai, Lek-Heng Lim
Stochastic optimization algorithms have become indispensable in machine learning.