We define discrete flat surfaces in hyperbolic 3-space from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean curvature 1 surfaces in hyperbolic 3-space, and we also describe discrete focal …
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48 results for “Discrete Space”
New discrete curves defined in space forms with geometric properties.
problem Defining discrete elastic and constrained elastic curves in space forms.
method Extending discrete Euclidean curvature to space forms and using Bäcklund transformations.
result Discrete elastic and constrained elastic curves are elements of a curve hierarchy.
Introduces new types of homogeneous spaces and their properties.
problem Defining and understanding new types of homogeneous spaces.
method Introducing and analyzing (strongly) (-)discrete homogeneous spaces.
result Discovers relationships between new and existing homogeneous space types.
In this paper, we consider the discrete deformation of the discrete space curves with constant torsion described by the discrete mKdV or the discrete sine-Gordon equations, and show that it is formulated as the torsion-preserving equidistant deformation on the osculating plane which satisfies the isoperimetric conditio…
No stable discrete maps into certain curved spaces exist.
problem Stability of discrete maps into curved spaces.
method Analysis of weighted length or energy functionals on graphs.
result Non-existence of stable discrete minimal immersions or harmonic maps into specific homogeneous spaces.
New proof for discrete Morse theory using combinatorial construction.
problem Verifying the Morse differential in discrete Morse homology.
method Combinatorial construction of flowlines in discrete Morse theory.
result Morse differential squares to zero in discrete Morse homology.
Unified framework extends adjoint Schrödinger bridge sampler to discrete spaces.
problem Challenges in learning discrete neural samplers due to gradients and combinatorial complexity.
method Introduces discrete ASBS, a unified framework that extends adjoint Schrödinger bridge sampler to discrete spaces.
result Empirically, discrete ASBS achieves competitive sample quality with significant advantages in training efficiency and scalability.
New methods create full discretized isothermic tori in Euclidean spaces.
problem Creating full discretized isothermic tori in Euclidean spaces.
method Using Darboux transformations and periodic curvature line systems.
result Discrete and semi-discrete k-dimensional isothermic tori in n-dimensional Euclidean space.
Short introduction to discrete flat fronts in hyperbolic space with a Weierstrass representation proof.
problem Understanding discrete flat fronts in hyperbolic space.
method Proving a Weierstrass representation for discrete flat fronts.
result Any discrete flat front in the mixed area sense admits a Weierstrass representation.
The paper finds formulas for special surface shapes in 3D space.
problem Creating formulas for constant mean curvature surfaces.
method Weierstrass representations for discrete surfaces in isotropic space.
result Constructs examples of surfaces with discrete parametrizations.
NES optimizes discrete structured VAEs effectively without gradient propagation.
problem Learning high-dimensional discrete latent spaces in generative models.
method Natural Evolution Strategies (NES) for gradient-free optimization of discrete structures.
result NES effectively optimizes discrete structured VAEs, comparable to gradient-based methods.
Gradient-based MCMC for discrete spaces improves sampling performance.
problem Sampling in discrete spaces using traditional methods is challenging.
method Introduced new discrete Metropolis-Hastings samplers inspired by MALA, with a novel preconditioning technique.
result Demonstrated strong empirical performance across various challenging sampling problems.
Study on Brownian motion on discrete curve spaces, proving stochastic completeness.
problem Analyzing Brownian motion on spaces of discrete curves.
method Introduced and studied Brownian motion on spaces of discrete regular curves with Sobolev-type metrics.
result All geodesically complete spaces of discrete regular curves are stochastically complete.
Survey on discrete minimal surfaces and their properties.
problem Discretizing minimal surfaces in Euclidean space.
method Polyhedral surfaces with parallel face offsets and circle patterns.
result All simply connected discrete minimal surfaces can be constructed from circle patterns.
Discrete diffusion samplers and bridges: Off-policy algorithms and applications in latent spacescs.LG
Discrete diffusion samplers improve sampling from unnormalised densities.
problem Sampling from discrete unnormalised densities efficiently.
method Introduce off-policy training techniques and data-to-energy Schrödinger bridge training for discrete diffusion samplers.
result Improved performance on synthetic and new benchmarks.
Using a quaternionic calculus, the Christoffel, Darboux, Goursat, and spectral transformations for discrete isothermic nets are described, with their interrelations. The Darboux and spectral transformations are used to define discrete analogs for cmc-1 surfaces in hyperbolic space and to obtain a discrete version of Br…
The local induction equation, or the binormal flow on space curves is a well-known model of deformation of space curves as it describes the dynamics of vortex filaments, and the complex curvature is governed by the nonlinear Schrödinger equation. In this paper, we present its discrete analogue, namely, a model of defor…
New RL algorithm tackles complex discrete action spaces.
problem Challenges in applying on-policy RL in high-dimensional discrete action spaces.
method Action-value critic, correlated actions, gradient sparsification.
result Empirically outperforms related on-policy algorithms.
Most existing deep reinforcement learning (DRL) frameworks consider either discrete action space or continuous action space solely. Motivated by applications in computer games, we consider the scenario with discrete-continuous hybrid action space. To handle hybrid action space, previous works either approximate the hyb…
Discretizes diffusions and harmonic functions on covering spaces.
problem Harmonic functions on covering spaces with bounded growth.
method Lyons-Sullivan discretizations of diffusion operators.
result Equivalence of discretized and continuous harmonic functions.
EDLP samples flat modes in discrete spaces using entropy.
problem Sampling flat modes in discrete spaces is challenging.
method EDLP uses a continuous auxiliary variable and local entropy to guide sampling.
result EDLP consistently outperforms traditional methods in various tasks.
Examines discrete curvature's relation to smooth curvature in 3 spaces.
problem Understanding how discrete curvature relates to smooth curvature in different spaces.
method Using specific triangular tilings of 3 types of spaces to examine curvatures.
result Discrete curvature can sense the smooth curvature of ambient space forms.
New algorithm estimates intrinsic dimension of discrete datasets.
problem Inaccuracies in using continuous methods for discrete datasets.
method Introduced an algorithm to infer intrinsic dimension of discrete spaces.
result Demonstrated accuracy on benchmark datasets and found a small intrinsic dimension in a metagenomic dataset.
We propose a unified definition for discrete analogues of constant mean curvature surfaces in spaces of constant curvature as a special case of discrete special isothermic nets. Bäcklund transformations and Lawson's correspondence are discussed. It is shown that the definition generalizes previous definitions and a con…
In this article, we sketch an algorithm that extends the Q-learning algorithms to the continuous action space domain. Our method is based on the discretization of the action space. Despite the commonly used discretization methods, our method does not increase the discretized problem dimensionality exponentially. We wil…
Adaptive discretization improves model-based RL in large spaces.
problem Efficient model-based reinforcement learning in large state-action spaces.
method Optimistic one-step value iteration with adaptive discretization.
result Adaptive discretization leads to better performance and lower memory usage.
Classifies graph configuration spaces homeomorphic to manifolds.
problem Classifying graph configuration spaces homeomorphic to manifolds.
method Developed techniques to translate topological properties into graph theoretic ones.
result Extended Abrams' work to classify certain graph configuration spaces.
A formula connects discrete harmonic surfaces to holomorphic functions.
problem Creating smooth discrete harmonic surfaces from holomorphic data.
method Weierstrass representation formula for discrete harmonic surfaces.
result Smooth converging sequence of discrete harmonic surfaces converges to a minimal surface.
CADD improves generative quality by augmenting discrete diffusion with continuous latent space.
problem Loss of semantic information between denoising steps in discrete diffusion models.
method Introduces a framework that augments discrete state space with a continuous latent space, allowing for graded, informative masked tokens.
result CADD improves generative quality across text generation, image synthesis, and code modeling.
Finding appropriate notions of discrete holomorphic maps and, more generally, conformal immersions of discrete Riemann surfaces into 3-space is an important problem of discrete differential geometry and computer visualization. We propose an approach to discrete conformality that is based on the concept of holomorphic l…
For Riemannian submersions, we establish some estimates for the spectrum of the total space in terms of the spectrum of the base space and the geometry of the fibers. In particular, for Riemannian submersions of complete manifolds with closed fibers of bounded mean curvature, we show that the spectrum of the base space…
Bäcklund transformations for smooth and ``space discrete'' Hashimoto surfaces are discussed and a geometric interpretation is given. It is shown that the complex curvature of a discrete space curve evolves with the discrete nonlinear Schrödinger equation (NLSE) of Ablowitz and Ladik, when the curve evolves with the Has…
Given a trivalent graph in the 3-dimensional Euclidean space, we call it a discrete surface because it has a tangent space at each vertex determined by its neighbor vertices. To abstract a continuum object hidden in the discrete surface, we introduce a subdivision method by applying the Goldberg-Coxeter subdivision and…
A new method trains discrete EBMs without sampling.
problem Training EBMs on discrete spaces is hard.
method Energy Discrepancy (ED), a contrastive loss.
result ED offers theoretical guarantees for various perturbation types.
We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of non-compact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor. In particular for rank one or simple Lie groups, Zariski dens…
New interpretation of discrete conformality using polyhedral convex hulls.
problem Understanding discrete conformality in 3D.
method Epstein-Penner convex hull construction and induced metrics.
result New bijections and interpretations of discrete conformality.
We answer affirmatively a question posed by Morita on homological stability of surface diffeomorphisms made discrete. In particular, we prove that -diffeomorphisms and volume preserving diffeomorphisms of surfaces as family of discrete groups exhibit homological stability. We show that the stable homology o…
The study determines discreteness of complex hyperbolic triangle groups.
problem Discreteness of complex hyperbolic triangle groups of specific type.
method Analysis of groups generated by complex reflections with given orders and distances.
result Determined intervals in parameter space for discrete and non-discrete groups.
Discrete linear Weingarten surfaces in space forms are characterized as special discrete -nets, a discrete analogue of Demoulin's -surfaces. It is shown that the Lie-geometric deformation of -nets descends to a Lawson transformation for discrete linear Weingarten surfaces, which coincides with the well-known L…
We give a Weierstrass type representation for semi-discrete minimal surfaces in Euclidean 3-space. We then give explicit parametrizations of various smooth, semi-discrete and fully-discrete catenoids, determined from either variational or integrable systems principles. Finally, we state the shared properties that those…
Calculates affine transformations for specific homogeneous spaces.
problem Computing groups of affine transformations on homogeneous spaces.
method Analyzes conditions for affine connections and uses them to establish group isomorphisms.
result Groups of affine transformations are locally isomorphic under specified conditions.
Rigidity theorem for discrete metric spaces embedded in Riemannian surfaces.
problem Understanding the rigidity of discrete metric spaces embedded in Riemannian surfaces.
method Proving that certain discrete metric spaces are rigidly embedded in the Euclidean plane or other Riemannian surfaces.
result Riemannian embeddings of certain discrete metric spaces are rigid, meaning they cannot be deformed without changing distances.
Paper solves POMDPs in continuous time and discrete spaces.
problem Optimal decision making in discrete state and action space systems under partial observability.
method Combining optimal filtering theory and deep learning to solve a Hamilton-Jacobi-Bellman equation.
result Derives a mathematical description and solution approach for continuous-time POMDPs.
Determining the space of free discrete two generator groups of Möbius transformations is an old and difficult problem. In this paper we show how to construct large balls of full dimension in this space. To do this, we begin with a marked discrete group of non-separating disjoint circle type. Such a group determines thr…
This primer explains diffusion models in general state spaces.
problem Diffusion models in general state spaces are not well-introduced.
method Develops discrete-time and continuous-time views of diffusion models, deriving Fokker-Planck and master equations.
result Unified understanding of diffusion models across continuous and discrete domains.
Proves existence of unique circle packings on polyhedral surfaces.
problem Existence of unique circle packings on polyhedral surfaces with specified discrete curvature.
method Constructs diffeomorphism between fiber bundles, uses discrete Ricci flow and edge flipping.
result Proves existence of unique inversive distance circle packings.
In this paper we introduce a discrete integrable system generalizing the discrete (real) cross-ratio system in to complex values of a generalized cross-ratio by considering as a real section of the complex Plücker quadric, realized as the space of two-spheres in We develop the geometry of the Plücker…
We consider a numerical approach for the incompressible surface Navier-Stokes equation. The approach is based on the covariant form and uses discrete exterior calculus (DEC) in space and a semi-implicit discretization in time. The discretization is described in detail and related to finite difference schemes on stagger…