相关论文: Learning on hexagonal structures and Monge-Amp\`er…

The Restricted Boltzmann Machine (RBM), an important tool used in machine learning in particular for unsupervized learning tasks, is investigated from the perspective of its spectral properties. Starting from empirical observations, we…

Deep generative models have become ubiquitous due to their ability to learn and sample from complex distributions. Despite the proliferation of various frameworks, the relationships among these models remain largely unexplored, a gap that…

Restricted Boltzmann machines (RBM) are generative models capable to learn data with a rich underlying structure. We study the teacher-student setting where a student RBM learns structured data generated by a teacher RBM. The amount of…

Humans excel at continually acquiring, consolidating, and retaining information from an ever-changing environment, whereas artificial neural networks (ANNs) exhibit catastrophic forgetting. There are considerable differences in the…

The growing volume of data produced by large astronomical surveys necessitates the development of efficient analysis techniques capable of effectively managing high-dimensional datasets. This study addresses this need by demonstrating some…

In this article, we extend the conventional framework of convolutional-Restricted-Boltzmann-Machine to learn highly abstract features among abitrary number of time related input maps by constructing a layer of multiplicative units, which…

We present necessary and sufficient conditions for a group homomorphism between spaces of smooth sections of Lie group bundles to be a weighted composition operator. These results provide new insights into a wide range of problems related…

Amphiphilic molecules spontaneously form self-assembled structures of various shapes depending on their molecular structures, the temperature, and other physical conditions. The functionalities of these structures are dictated by their…

We review recent advances in the numerical analysis of the Monge-Amp\`ere equation. Various computational techniques are discussed including wide-stencil finite difference schemes, two-scaled methods, finite element methods, and methods…

The manifold hypothesis states that high-dimensional data can be modeled as lying on or near a low-dimensional, nonlinear manifold. Variational Autoencoders (VAEs) approximate this manifold by learning mappings from low-dimensional latent…

Manifold learning is a popular and quickly-growing subfield of machine learning based on the assumption that one's observed data lie on a low-dimensional manifold embedded in a higher-dimensional space. This thesis presents a mathematical…

In this paper we consider a fractional analogue of the Monge-Amp\`ere operator. Our operator is a concave envelope of fractional linear operators of the form $ \inf_{A\in \mathcal{A}}L_Au, $ where the set of operators corresponds to all…

An obstacle to artificial general intelligence is set by continual learning of multiple tasks of different nature. Recently, various heuristic tricks, both from machine learning and from neuroscience angles, were proposed, but they lack a…

We introduce a machine-learning approach for identifying hidden structural features of open quantum dynamics under restricted experimental access. Unlike most existing data-driven methods which focus on detection or prediction of dynamical…

The Boltzmann Machine (BM) is a neural network composed of stochastically firing neurons that can learn complex probability distributions by adapting the synaptic interactions between the neurons. BMs represent a very generic class of…

Disentanglement is a useful property in representation learning which increases the interpretability of generative models such as Variational autoencoders (VAE), Generative Adversarial Models, and their many variants. Typically in such…

In this paper, we establish local and global regularity estimates for linearized Monge-Amp\`ere equations in divergence form via critical Lorentz space estimates for the Green's function of the linearized Monge-Amp\`ere operator and its…

Let $X$ be a compact complex manifold which admits a hermitian metric satisfying a curvature condition introduced by Guan-Li. Given a semipositive form $\theta$ with positive volume, we define the Monge-Amp\`ere operator for unbounded…

We introduce a random matrix framework for studying statistical-mechanical lattice systems through spectral observables. Equilibrium configurations sampled from a Boltzmann measure are mapped to matrix ensembles whose covariance structure…

We prove that every $\mathcal{C}^1(\bar\omega)$-regular subsolution of the Monge-Amp\`ere system posed on a $2$-dimensional domain $\omega$ and with target codimension $2$, can be uniformly approximated by its exact solutions with…