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

We construct solutions to Monge-Amp\`ere equations whose Monge-Amp\`ere measures contain singular components supported on low codimensional sets. We also study the regularity of such solutions. To motivate our construction, we present…

Recently, there has been much interest in spectral approaches to learning manifolds---so-called kernel eigenmap methods. These methods have had some successes, but their applicability is limited because they are not robust to noise. To…

Predicting the structure of multi-protein complexes is a grand challenge in biochemistry, with major implications for basic science and drug discovery. Computational structure prediction methods generally leverage pre-defined structural…

Statistical learning algorithms are finding more and more applications in science and technology. Atomic-scale modeling is no exception, with machine learning becoming commonplace as a tool to predict energy, forces and properties of…

Monge-Amp\`ere equations of the form, $u_{xx}u_{yy}-u_{xy}^2=F(u,u_x,u_y)$ arise in many areas of fluid and solid mechanics. Here it is shown that in the special case $F=u_y^4f(u, u_x/u_y)$, where $f$ denotes an arbitrary function, the…

This is an expository paper describing how duality theory for Hessian manifolds provides a natural setting for optimal transport. We explain how this can be used to solve Monge-Amp\`ere equations and survey recent results along these lines…

For four decades statistical physics has been providing a framework to analyse neural networks. A long-standing question remained on its capacity to tackle deep learning models capturing rich feature learning effects, thus going beyond the…

Humans are extremely swift learners. We are able to grasp highly abstract notions, whether they come from art perception or pure mathematics. Current machine learning techniques demonstrate astonishing results in extracting patterns in…

We construct convex functions on $\mathbb{R}^3$ and $\mathbb{R}^4$ that are smooth solutions to the Monge-Amp\`{e}re equation $\det D^2u = 1$ away from compact one-dimensional singular sets, which can be Y-shaped or form the edges of a…

In this paper we extend our previous work on singularities of Monge-Amp\`ere foliations to the case of pseudoconvex finite type domains. We are able to answer the questin of Burns on homogeneous polynomials whose logarithm satisfies the…

Statistical learning in high-dimensional spaces is challenging without a strong underlying data structure. Recent advances with foundational models suggest that text and image data contain such hidden structures, which help mitigate the…

We study asymptotic behaviors of solutions to the Monge-Amp\`ere equation in cones and use the expansion as a tool to study the regularity of solutions in polygonal domains.

The manifold hypothesis (MH) is often used to explain how machine learning can overcome the curse of dimensionality. However, the MH is only applicable in regimes where the training data provides a sufficiently dense sample of the…

We propose a heterogeneous graph mamba network (HGMN) as the first exploration in leveraging the selective state space models (SSSMs) for heterogeneous graph learning. Compared with the literature, our HGMN overcomes two major challenges:…

We investigate whether quantum annealers with select chip layouts can outperform classical computers in reinforcement learning tasks. We associate a transverse field Ising spin Hamiltonian with a layout of qubits similar to that of a deep…

A novel extension of the Probabilistic Learning on Manifolds (PLoM) is presented. It makes it possible to synthesize solutions to a wide range of nonlinear stochastic boundary value problems described by partial differential equations…

We introduce two constructions in geometric deep learning for 1) transporting orientation-dependent convolutional filters over a manifold in a continuous way and thereby defining a convolution operator that naturally incorporates the…

The continuous effort towards topological quantum devices calls for an efficient and non-invasive method to assess the conformity of components in different topological phases. Here, we show that machine learning paves the way towards…

We introduce machine learning methodology to the study of lattice polytopes. With supervised learning techniques, we predict standard properties such as volume, dual volume, reflexivity, etc, with accuracies up to 100%. We focus on 2d…

Automata learning has been successfully applied in the verification of hardware and software. The size of the automaton model learned is a bottleneck for scalability, and hence optimizations that enable learning of compact representations…