Search Results for author: Lek-Heng Lim

Found 9 papers, 2 papers with code

What is an equivariant neural network?

no code implementations15 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.

Protein Structure Prediction

Recht-Ré Noncommutative Arithmetic-Geometric Mean Conjecture is False

no code implementations2 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 $n = 5$.

Stochastic Optimization

Topology of deep neural networks

no code implementations13 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 ($\approx 0. 01\%$).

Best k-layer neural network approximations

no code implementations2 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.

Semi-Riemannian Manifold Optimization

1 code implementation18 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

Tropical Geometry of Deep Neural Networks

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.

Cohomology of Cryo-Electron Microscopy

no code implementations5 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.

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