Related papers: Learning on hexagonal structures and Monge-Amp\`er…

In this paper we correct a gap of Whyburn type topological lemma and establish two superior limit theorems. As the applications of our Whyburn type topological theorems, we study the following Monge-Amp\`{e}re equation \begin{eqnarray}…

For three decades statistical mechanics has been providing a framework to analyse neural networks. However, the theoretically tractable models, e.g., perceptrons, random features models and kernel machines, or multi-index models and…

We establish global H\"older estimates for solutions to inhomogeneous linearized Monge-Amp\`ere equations in two dimensions with the right hand side being the divergence of a bounded vector field. These equations arise in the…

The real homogeneous Monge-Amp\`{e}re equation in one space and one time dimensions admits infinitely many Hamiltonian operators and is completely integrable by Magri's theorem. This remarkable property holds in arbitrary number of…

We develop information geometric techniques to understand the representations learned by deep networks when they are trained on different tasks using supervised, meta-, semi-supervised and contrastive learning. We shed light on the…

The manifold hypothesis posits that high-dimensional data typically resides on low-dimensional sub spaces. In this paper, we assume manifold hypothesis to investigate graph-based semi-supervised learning methods. In particular, we examine…

A Machine can only learn if it is biased in some way. Typically the bias is supplied by hand, for example through the choice of an appropriate set of features. However, if the learning machine is embedded within an {\em environment} of…

The requirement of generating predictions that exactly fulfill the fundamental symmetry of the corresponding physical quantities has profoundly shaped the development of machine-learning models for physical simulations. In many cases,…

Deep Boltzmann machines (DBMs), one of the first ``deep'' learning methods ever studied, are multi-layered probabilistic models governed by a pairwise energy function that describes the likelihood of all variables/nodes in the network. In…

Unsupervised learning in a generalized Hopfield associative-memory network is investigated in this work. First, we prove that the (generalized) Hopfield model is equivalent to a semi-restricted Boltzmann machine with a layer of visible…

Bayesian networks faithfully represent the symmetric conditional independences existing between the components of a random vector. Staged trees are an extension of Bayesian networks for categorical random vectors whose graph represents…

We study the problem of estimating a manifold from random samples. In particular, we consider piecewise constant and piecewise linear estimators induced by k-means and k-flats, and analyze their performance. We extend previous results for…

Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data…

Understanding the reasons for the success of deep neural networks trained using stochastic gradient-based methods is a key open problem for the nascent theory of deep learning. The types of data where these networks are most successful,…

We consider a group of strategic agents who must each repeatedly take one of two possible actions. They learn which of the two actions is preferable from initial private signals, and by observing the actions of their neighbors in a social…

The idea that data lies on a non-linear space has brought up the concept of manifold learning as a part of machine learning.

Restricted Boltzmann Machines (RBMs) offer a versatile architecture for unsupervised machine learning that can in principle approximate any target probability distribution with arbitrary accuracy. However, the RBM model is usually not…

Traditional manifold learning algorithms assumed that the embedded manifold is globally or locally isometric to Euclidean space. Under this assumption, they divided manifold into a set of overlapping local patches which are locally…

Using Monge-Amp\`ere geometry, we study the singular structure of a class of nonlinear Monge-Amp\`ere equations in three dimensions, arising in geophysical fluid dynamics. We extend seminal earlier work on Monge-Amp\`ere geometry by…

Processes are often viewed as coalgebras, with the structure maps specifying the state transitions. In the simplest case, the state spaces are discrete, and the structure map simply takes each state to the next states. But the coalgebraic…