We introduce the discrete Einstein metrics as critical points of discrete energy on triangulated 3-manifolds, and study them by discrete curvature flow of second (fourth) order. We also study the convergence of the discrete curvature flow. Discrete curvature flow of second order is an analogue of smooth Ricci flow.
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48 results for “discrete flow”
Paper proposes a new generative model for discrete distributions using flows on submanifolds.
problem Discretization issues and complex statistical dependencies in discrete data.
method Continuous normalizing flows on factorizing discrete measures, geodesic flow matching.
result Efficient training and broad applicability demonstrated through experiments.
While normalizing flows have led to significant advances in modeling high-dimensional continuous distributions, their applicability to discrete distributions remains unknown. In this paper, we show that flows can in fact be extended to discrete events---and under a simple change-of-variables formula not requiring log-d…
Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, Imath.GT
In this paper, we introduce a parameterized discrete curvature (-curvature) for piecewise linear metrics on polyhedral surfaces, which is a generalization of the classical discrete curvature. A discrete uniformization theorem is established for the parameterized discrete curvature, which generalizes the discrete uni…
Unified framework for continuous-state discrete flow matching models.
problem Discrete generative modeling with continuous probabilities.
method Introducing -Flow, a family of CS-DFM models based on information geometry.
result Optimal flow matching loss for -flow minimizes generalized kinetic energy.
Study of Ricci flow on discrete surfaces of revolution with constant Gaussian curvature.
problem Understanding Ricci flow on discrete surfaces of revolution.
method Explicit parametrizations and Ricci flow analysis for discrete surfaces of revolution.
result Discrete surfaces of revolution approach constant Gaussian curvature under Ricci flow.
IDF++ improves integer discrete flows for lossless compression.
problem Theoretical limitations of integer discrete flows for lossless compression.
method Investigated and improved integer discrete flows, addressing gradient bias and architecture modifications.
result Different architecture modifications improve integer discrete flows for lossless compression.
Discrete Flow Mapsstat.ML
Discrete Flow Maps bypass sequential prediction limits for parallel text generation.
problem Sequential autoregressive prediction limits large language model speed.
method Flow Maps compress generative trajectories into single-step mappings.
result Discrete Flow Maps surpass previous state-of-the-art results in discrete flow modeling.
DFMs enable flow-based models for multimodal discrete and continuous data.
problem Combining discrete and continuous data for generative models.
method Discrete Flow Models (DFMs) using Continuous Time Markov Chains.
result DFMs achieve state-of-the-art co-design performance for protein structure and sequence generation.
The paper studies deformation of discrete conformal structures on surfaces using combinatorial curvature flows.
problem Finding piecewise constant curvature metrics on surfaces with prescribed combinatorial curvatures.
method Combinatorial curvature flows, including Ricci flow and Calabi flow, are applied to deform Glickenstein's discrete conformal structures.
result The solution of the combinatorial Ricci flow can be uniquely extended and converges exponentially fast for any initial value under certain conditions.
Develops a new method for learning discrete distributions without embedding them in a continuous space.
problem Challenges in learning discrete distributions using current methodologies.
method Introduces a MAD invertible map and a mixed variational flow (MAD Mix) for discrete distributions.
result MAD Mix produces more reliable approximations than continuous-embedding flows.
New method reduces discrete flow transitions, improving perplexity estimation.
problem Stochasticity in discrete paths makes rectification strategies ineffective.
method Dynamic-optimal-transport-like minimization objective with minibatch strategies.
result 32 times reduction in transitions for same perplexity.
Fractional combinatorial flow improves surface conformal structures.
problem Improving discrete conformal structures on surfaces.
method Introducing a fractional combinatorial Calabi flow for discrete conformal structures on surfaces.
result Longtime existence and global convergence of the fractional combinatorial Calabi flow for various surface types.
We define Discrete Quasi-Einstein metrics (DQE-metrics) as the critical points of discrete total curvature functional on triangulated 3-manifolds. We study DQE-metrics by introducing some combinatorial curvature flows. We prove that these flows produce solutions which converge to discrete quasi-Einstein metrics when th…
Study on deforming discrete conformal structures on surfaces with boundaries.
problem Deforming discrete conformal structures on surfaces with boundaries.
method Introduce combinatorial Ricci flow and combinatorial Calabi flow, establish longtime existence and global convergence of solutions.
result Effective algorithms for finding discrete hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.
The paper introduces combinatorial curvature and flow for polyhedral surfaces, proving rigidity and solving the Yamabe problem.
problem Discrete conformal structures on polyhedral surfaces and their rigidity.
method Parameterized combinatorial curvature, combinatorial α-Ricci flow, and flow extension through singularities.
result Existence and convergence of combinatorial α-Ricci flow for solving the Yamabe problem.
Unified error analysis for discrete flow models.
problem Error analysis of discrete flow models.
method Stochastic calculus theory, Girsanov theorem, generator matching, uniformization.
result First error analysis for discrete flow models.
The paper establishes a discrete uniformization theorem for surfaces with piecewise hyperbolic metrics.
problem Finding decorated piecewise hyperbolic metrics with prescribed combinatorial curvature.
method Introduced combinatorial α-Ricci flow with surgery to handle potential singularities and prove longtime existence and convergence.
result Existence of decorated piecewise hyperbolic metrics with prescribed combinatorial α-curvature.
The paper studies stability of discretized Anosov flows.
problem Global stability of discretized Anosov flows.
method Defined and proved equivalence with previous definitions, showed properties through openness and closedness, and established integrability and uniqueness of invariant foliations.
result Discretized Anosov flows are globally stable.
New discrete conformal structures on surfaces with boundary, proving global rigidity and constructing hyperbolic metrics.
problem Creating new discrete conformal structures on surfaces with boundary.
method Introducing new discrete conformal structures, proving global rigidity using variational principles, and introducing combinatorial curvature flows.
result Global rigidity of new discrete conformal structures and effective algorithms for constructing hyperbolic metrics.
Flow models have recently made great progress at modeling ordinal discrete data such as images and audio. Due to the continuous nature of flow models, dequantization is typically applied when using them for such discrete data, resulting in lower bound estimates of the likelihood. In this paper, we introduce subset flow…
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…
Normalizing flows are a powerful class of generative models for continuous random variables, showing both strong model flexibility and the potential for non-autoregressive generation. These benefits are also desired when modeling discrete random variables such as text, but directly applying normalizing flows to discret…
In this paper, we study the discrete Morse flow for the Ricci flow on football, which is the 2-sphere with removed north and south poles and with the metric of constant scalar curvature, and and for Porous media equation on a bounded regular domain in the plane. We show that with a suitable assumption about $g(0)…
The paper studies the consistency of mean curvature flow via volumetric varifolds.
problem Consistency of mean curvature flow.
method Discretization using volumetric varifolds and derivation of Brakke approximate equality.
result Derivation of a Brakke approximate equality involving varifold masses and approximate mean curvatures.
Corrected samplers reduce discretization error in discrete flow models without additional computational cost.
problem Discretization error in samplers for discrete flow models.
method Established non-asymptotic error bounds for samplers, proposed time-corrected and location-corrected samplers.
result Location-corrected sampler has lower complexity and better generation quality.
Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, IImath.GT
The paper develops algorithms for finding metrics with prescribed combinatorial curvature on polyhedral surfaces.
problem Finding metrics with prescribed combinatorial curvature on polyhedral surfaces.
method Discrete uniformization theorem, combinatorial α-Yamabe flow, combinatorial α-Calabi flow, edge flipping surgery.
result Longtime existence and convergence of combinatorial α-Yamabe flow and combinatorial α-Calabi flow with surgery.
Exact discrete mechanics for nonholonomic systems defined.
problem Discrete mechanics for nonholonomic systems.
method Constructing an exponential map and deriving exact discrete nonholonomic integrators.
result Reproduces continuous nonholonomic flow as discrete flow on constraint submanifold.
Improved explicit estimates for the discrete Laplace operator with hyperbolic circle patternsmath.DG
Paper improves estimates for discrete Laplace in hyperbolic geometry.
problem Establishing compactness for discrete Laplace in hyperbolic geometry.
method Explicit estimates for discrete Laplace based on Glickenstein-Thomas formulation.
result New proofs of long time existence for Calabi flows in hyperbolic geometry.
In this paper we investigate flows on discrete curves in $\C^2$, $\CP^1$, and $\C$. A novel interpretation of the one dimensional Toda lattice hierarchy and reductions thereof as flows on discrete curves will be given.
A discrete method approximates hyperbolic curvature flow in the plane.
problem Modeling wave phenomena in solid-liquid interfaces.
method Semidiscrete finite difference method for hyperbolic curvature flow.
result Error bounds for natural discrete norms are proven.
A new method for categorical variational inference using discrete normalizing flows.
problem Challenges in optimizing variational approximations for discrete latent variables.
method Differentiable reparameterization using a mixture of discrete normalizing flows.
result Improves optimization of evidence lower bound and reduces sensitivity to hyperparameters.
Forward-Euler fails for simulating Wasserstein gradient flows with KL divergence.
problem Simulating Wasserstein gradient flows with forward-Euler discretization fails for KL divergence.
method Forward-Euler discretization for Wasserstein gradient flows with KL divergence.
result Forward-Euler discretization can be incorrect for Wasserstein gradient flows with KL divergence.
Flow based models such as Real NVP are an extremely powerful approach to density estimation. However, existing flow based models are restricted to transforming continuous densities over a continuous input space into similarly continuous distributions over continuous latent variables. This makes them poorly suited for m…
Neural networks' feature geometry evolves like discrete Ricci flow.
problem Understanding neural feature representations and their geometric transformations.
method Approximating input manifold with geometric graphs and analyzing their evolution during training.
result Neural feature geometry evolves like discrete Ricci flow, with nonlinear activations playing a crucial role.
A new method for discrete data normalizing flows using latent transformations.
problem Challenges in parameterizing bijective transformations for discrete data.
method Predict a distribution over latent transformations to make the marginal likelihood differentiable.
result Discrete-data normalizing flows can be trained using gradient-based learning with unbiased score function estimation.
New method learns discrete graph diffusion via free-energy gradient flows.
problem Challenges in translating continuous diffusion models to discrete spaces.
method Proposes a novel computational approach using a specific metric on the simplex.
result Recover the underlying functional for various graph classes.
We present a notion of super Ricci flow for time-dependent finite weighted graphs. A challenging feature is that these flows typically encounter singularities where the underlying graph structure changes. Our notion is robust enough to allow the flow to continue past these singularities. As a crucial tool for this purp…
Improved sampling quality with RBM-Flow and D-Flow models.
problem Efficient sampling of complex data distributions using invertible flows.
method Implement RBM-Flow and D-Flow models with discrete latent variables.
result Significant improvement in sampling quality over baseline models.
Branching Flows generates sequences of varying lengths using binary trees.
problem Generating sequences of unknown lengths or fixed elements.
method A generative modeling framework that evolves states over binary trees, controlling sequence length.
result Branching Flows can generate sequences of varying lengths and mix different types of state spaces.
Discretely Indexed Flowsstat.ML
DIF extends NF with stochastic discrete latent variables for better density estimation.
problem Improving density estimation with discontinuities and fine details.
method Discretely indexed flows as an extension of Normalizing Flows with stochastic latent variables.
result DIF inherit good computational behavior of NF and can capture distributions with discontinuities.
Lossless compression methods shorten the expected representation size of data without loss of information, using a statistical model. Flow-based models are attractive in this setting because they admit exact likelihood optimization, which is equivalent to minimizing the expected number of bits per message. However, con…
Generative Assignment Flows for Representing and Learning Joint Distributions of Discrete Datastat.ML
Generative model for joint discrete distributions using randomized assignment flows.
problem Efficiently representing and sampling from complex joint distributions of discrete variables.
method Randomized assignment flows on the statistical submanifold of factorizing distributions.
result Our model can efficiently represent and sample from any target distribution and assess likelihood of unseen data points.
The elastic flow, which is the -gradient flow of the elastic energy, has several applications in geometry and elasticity theory. We present stable discretizations for the elastic flow in two-dimensional Riemannian manifolds that are conformally flat, i.e.\ conformally equivalent to the Euclidean space. Examples in…
ReDi improves few-step generation for discrete data models.
problem Slow sampling speeds in discrete flow-based models.
method Rectified Discrete Flow (ReDi) reduces factorization error by rectifying coupling.
result Empirically, ReDi reduces Conditional Total Correlation and enables few-step generation.
Ricci flow deforms the Riemannian metric proportionally to the curvature, such that the curvature evolves according to a heat diffusion process and eventually becomes constant everywhere. Ricci flow has demonstrated its great potential by solving various problems in many fields, which can be hardly handled by alternati…
This work considers the question of whether mean-curvature flow can be modified to avoid the formation of singularities. We analyze the finite-elements discretization and demonstrate why the original flow can result in numerical instability due to division by zero. We propose a variation on the flow that removes the nu…
Study combinatorial Yamabe flow on infinite triangulated surfaces.
problem Solve discrete Yamabe problem on noncompact surfaces.
method Introduced and analyzed combinatorial Yamabe flow with short-time and long-time existence proofs.
result Established short-time and long-time existence of the flow, and proved convergence in hexagonal triangulations.