Related papers: Quantum Geometry insights in Deep Learning
We construct a special plurisubharmonic defining function for a smoothly bounded strictly pseudoconvex domain so that the determinant of the complex Hessian vanishes to high order on the boundary. This construction, coupled with regularity…
The current work addresses quantum machine learning in the context of Quantum Artificial Neural Networks such that the networks' processing is divided in two stages: the learning stage, where the network converges to a specific quantum…
The inverse reflector problem arises in geometrical nonimaging optics: Given a light source and a target, the question is how to design a reflecting free-form surface such that a desired light density distribution is generated on the…
There has been much work in the recent past in developing the idea of quantum geometry to characterize and understand the structure of many-particle states. For mean-field states, the quantum geometry has been defined and analysed in terms…
We investigate a class of multi-dimensional two-component systems of Monge-Amp\`ere type that can be viewed as generalisations of heavenly-type equations appearing in self-dual Ricci-flat geometry. Based on the Jordan-Kronecker theory of…
Auto-encoders are perhaps the best-known non-probabilistic methods for representation learning. They are conceptually simple and easy to train. Recent theoretical work has shed light on their ability to capture manifold structure, and drawn…
Recently the general form of a translation-covariant quantum Boltzmann equation has been derived which describes the dynamics of a tracer particle in a quantum gas. We develop a stochastic wave function algorithm that enables full…
Optimal Transport is a foundational mathematical theory that connects optimization, partial differential equations, and probability. It offers a powerful framework for comparing probability distributions and has recently become an important…
Neural networks (NNs) representing quantum states are typically trained using Markov chain Monte Carlo based methods. However, unless specifically designed, such samplers only consist of local moves, making the slow-mixing problem prominent…
The ultimate aim of the study is to explore the inverse design of porous metamaterials using a deep learning-based generative framework. Specifically, we develop a property-variational autoencoder (pVAE), a variational autoencoder (VAE)…
Multi-Agent Reinforcement Learning involves agents that learn together in a shared environment, leading to emergent dynamics sensitive to initial conditions and parameter variations. A Dynamical Systems approach, which studies the evolution…
The promise of quantum neural nets, which utilize quantum effects to model complex data sets, has made their development an aspirational goal for quantum machine learning and quantum computing in general. Here we provide new methods of…
Optimal transport and information geometry both study geometric structures on spaces of probability distributions. Optimal transport characterizes the cost-minimizing movement from one distribution to another, while information geometry…
Quantum computing holds great potential for advancing the limitations of machine learning algorithms to handle higher dimensions of data and reduce overall training parameters in deep learning (DL) models. This study uses a trainable…
In recent years, deep learning has had a profound impact on machine learning and artificial intelligence. At the same time, algorithms for quantum computers have been shown to efficiently solve some problems that are intractable on…
Modern deep learning has enabled unprecedented achievements in various domains. Nonetheless, employment of machine learning for wave function representations is focused on more traditional architectures such as restricted Boltzmann machines…
Given a quantum circuit, a quantum computer can sample the output distribution exponentially faster in the number of bits than classical computers. A similar exponential separation has yet to be established in generative models through…
We study the regularity and the growth rates of solutions to two-dimensional Monge-Amp\`ere equations with the right-hand side exhibiting polynomial growth. Utilizing this analysis, we demonstrate that the translators for the flow by…
An equation of Monge-Amp\`ere type has, for the first time, been solved numerically on the surface of the sphere in order to generate optimally transported (OT) meshes, equidistributed with respect to a monitor function. Optimal transport…
Determining the absolute configuration of gas-phase molecules in position-space has long been a fundamental challenge in molecular physics. While strong-field-induced Coulomb explosion imaging (CEI) has emerged as a powerful tool for…