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Found 11 papers, 2 papers with code
Cohomology of Cryo-Electron Microscopy
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.
Tropical Geometry of Deep Neural Networks
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.
Semi-Riemannian Manifold Optimization
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
Best k-layer neural network approximations
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.
Topology of deep neural networks
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 ($\approx 0. 01\%$).
Recht-Ré Noncommutative Arithmetic-Geometric Mean Conjecture is False
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 $n = 5$.
What is an equivariant neural network?
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.
LU decomposition and Toeplitz decomposition of a neural network
It is well-known that any matrix $A$ has an LU decomposition.
Stochastic Steffensen method
Is it possible for a first-order method, i. e., only first derivatives allowed, to be quadratically convergent?
Recht-Re Noncommutative Arithmetic-Geometric Mean Conjecture is False
Stochastic optimization algorithms have become indispensable in machine learning.
Recht-Re Noncommutative Arithmetic-Geometric Mean Conjecture is False
Stochastic optimization algorithms have become indispensable in machine learning.
