Study shows continuous evolution of curves in Fréchet distance.
problem Continuous evolution of curves under curvature flow.
method Curvature flow and level-set flow, analyzed in Fréchet distance.
result Evolution of curves depends continuously on initial curve.
Study geometric flows with varying parameters and prove continuous dependence.
problem Continuous dependence of flows on parameters in geometric settings.
method Derived suitable topologies for vector fields and flows, proved new continuous dependence.
result Proved continuous dependence of flows on parameters in a general topological space.
Continuity of earthquake flow map transfers Teichmüller dynamics results.
problem Transfer results from Teichmüller dynamics to earthquake flow.
method Analyze continuity of earthquake flow map and its inverse.
result Transfer results from Teichmüller dynamics to earthquake flow.
Study on Gaussian interpolation flows for generative modeling.
problem Theoretical properties and regularizing effect of Gaussian denoising in continuous normalizing flows.
method Unified framework of Gaussian interpolation flow, Lipschitz regularity, existence and uniqueness of flow, stability analysis.
result Established theoretical properties of Gaussian interpolation flows, including Lipschitz continuity and existence of flow.
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.
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.
Defines mass for non-smooth hyperbolic spaces using a modified flow.
problem Defining mass for non-smooth, asymptotically hyperbolic spaces.
method Normalized Ricci-DeTurck flow with scalar curvature lower bound.
result Mass function well-defined for continuous metrics.
Study shows uniform decay rate for singular mean curvature flows.
problem Understanding singularities in mean curvature flows.
method Rescaled flow analysis near compact singularities.
result Uniform decay order bound for the rescaled flow.
Continuous curve evolution depends on initial shape on sphere.
problem Evolution of a curve on a sphere by curvature flow.
method Study of curve evolution using curvature flow and level-set flow.
result Evolution depends continuously on initial curve in Fréchet distance.
Uniform proof for Ricci flows on complete manifolds.
problem Proving short-time existence, uniqueness, and continuous dependence for Ricci flows.
method Using Koch-Lamm framework and tensor heat kernel estimates, with a new continuous dependence estimate.
result Uniform proof of short-time existence, uniqueness, and continuous dependence for Ricci flows.
Strict type-II blowup in harmonic map flow is proven to have Hölder continuous body map.
problem Finite-time singularity of harmonic map flow.
method Analysis of outer energy scale and Hölder continuity proof.
result Strictly type-II blowup body map is Hölder continuous.
In this article, we consider the Angenent-Caputo-Knopf's Ricci Flow through neckpinch singularities. We will explain how one can see the A-C-K's Ricci flow through a neckpinch singularity as a flow of integral current spaces. We then prove the continuity of this weak flow with respect to the Sormani-Wenger Intrinsic Fl…
Study shows continuity and geometric regularity of Kähler-Ricci flow blow-up limits.
problem Geometric regularity of blow-up limits of the Kähler-Ricci flow.
method Established geometric regularity for Type I blow-up limits based on sequences of Ricci vertices.
result The limiting flow is continuous in time in Gromov-Hausdorff and Gromov-W1 distance. Paper introduces Categorical Normalizing Flows for better handling of categorical data.
problem Limited application of normalizing flows on categorical data due to lack of intrinsic order.
method Categorical Normalizing Flows use continuous transformations to model latent relations in categorical data, optimizing both continuous representation and model likelihood.
result GraphCNF, a permutation-invariant generative model, outperforms state-of-the-art on molecule generation.
We show that normalising flows become pathological when used to model targets whose supports have complicated topologies. In this scenario, we prove that a flow must become arbitrarily numerically noninvertible in order to approximate the target closely. This result has implications for all flow-based models, and espec…
Characterizes billiard and quasigeodesic flows in polyhedral convex bodies.
problem Characterizing billiard and quasigeodesic flows in polyhedral convex bodies.
method Alexandrov geometry methods.
result Optimal regularity result for convex bodies: billiard dynamics is continuous if boundary is of class C2,1. 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…
New path-gradient estimator for continuous normalizing flows.
problem Limitation of simple Gaussian variational distributions in complex applications.
method Proposed a path-gradient estimator for continuous normalizing flows.
result Empirical evidence of superior performance of the new estimator.
Study on curvature blow-up and convergence of continuity method on Hirzebruch surface.
problem Curvature blow-up and convergence of continuity method on Hirzebruch surface.
method Continuity method applied to generalised Hirzebruch surface, focusing on Gromov-Hausdorff convergence and scalar curvature estimates.
result A general solution to the continuity method either exists or all times, or the scalar curvature blows up.
Unified framework for continuous-state discrete flow matching models.
problem Discrete generative modeling with continuous probabilities.
method Introducing α-Flow, a family of CS-DFM models based on information geometry. result Optimal flow matching loss for α-flow minimizes generalized kinetic energy. New method trains normalizing flows using entropy-regularized transport.
problem Training continuous normalizing flows efficiently.
method Formulates flows as gradients of scalar potentials, training only these potentials.
result Trains normalizing flows without explicit flow computation during training.
CT-OT Flow estimates continuous-time dynamics from discrete snapshots.
problem Estimating continuous-time dynamics from temporally aggregated snapshots with noisy or uncertain timestamps.
method Two-stage framework: aligning neighboring intervals via partial optimal transport (POT) and reconstructing a continuous-time distribution through temporal kernel smoothing.
result Reduces distributional and trajectory errors compared with existing methods across synthetic and real datasets.
We show that the isoperimetric profile hg(t)(ξ) of a compact Riemannian manifold (M,g) is jointly continuous when metrics g(t) vary continuously. We also show that, when M is a compact surface and g(t) evolves under normalized Ricci flow, hg(t)2(ξ) is uniform Lipschitz continuous and hence $h_{g(t)}(…
EWFM trains continuous flows with only energy evaluations, improving sample quality with fewer computations.
problem Efficiently sampling from complex, high-dimensional Boltzmann distributions using only energy evaluations.
method Energy-Weighted Flow Matching (EWFM) using importance sampling and iterative/annealed training.
result Improved sample quality with up to 3 orders of magnitude fewer energy evaluations compared to existing methods.
Continuous time analysis of bubble formation in harmonic maps.
problem Understanding bubble formation in harmonic map heat flow.
method Continuous time approach to analyze bubbling sequences.
result Solutions approach multi-bubble configurations in continuous time.
Neural Manifold ODEs improve manifold data modeling.
problem Adapting deep generative models to non-Euclidean spaces.
method Introducing Neural Manifold ODEs for manifold generalization and continuous probability computation.
result Improves density estimation and downstream tasks on arbitrary manifolds.
CLPF models continuous time-series data with improved representational power and variational approximations.
problem Fitting continuous time-series data with existing models faces challenges in representational power and variational quality.
method CLPF uses a time-dependent normalizing flow driven by a stochastic differential equation to decode continuous latent processes into continuous observables. Maximum likelihood optimization is achieved through a novel variational posterior process.
result CLPF outperforms state-of-the-art baselines on synthetic and real-world time-series data.
Hybrid model avoids forgetting across tasks and classes.
problem Challenges of continual learning in modern deep learning.
method Hybrid generative-discriminative approach using normalizing flows.
result Strong performance on various continual learning benchmarks.
Proposes a continuous flow model to understand and control instability in gradient descent for deep learning.
problem Understanding and controlling the instability of gradient descent in deep learning.
method Introduces the Principal Flow (PF), a continuous time flow that approximates gradient descent dynamics.
result The PF captures divergent and oscillatory behaviors of gradient descent, including escaping local minima and saddle points.
Geometric correspondence links flow metrics to reparameterizations.
problem Linking flow metrics to reparameterizations of geodesic flows.
method Analysis of Mineyev's flow space and Green metrics.
result First examples of continuous reparameterizations on negatively curved manifolds.
Quantum computers can simulate flow models efficiently.
problem Efficiently simulating continuous flow models on quantum computers.
method Relating flow models to the Schrödinger equation and proving efficient Hamiltonian simulation.
result Quantum computers can prepare qsamples for flow models efficiently.
Harmonic map flow preserves almost-holomorphic maps without singularities.
problem Preserving almost-holomorphic maps without singularities.
method Harmonic map flow applied to almost-holomorphic maps.
result No singularities or necks appear in the limit at singular times.
Proves simplicity of Lyapunov exponents for specific Anosov flows.
problem Proving all Lyapunov exponents have multiplicity 1 for certain Anosov flows.
method Perturbative results for flows, modification of eigenvalues, Markov partition, and simplicity criterion.
result In a C1-open and Ck-dense set of Anosov flows, all Lyapunov exponents have multiplicity 1. Existence and uniqueness theorem for Ricci flow on weighted graphs proved.
problem Existence and uniqueness of solutions to Ricci flow equations on weighted graphs.
method Continuous time normalized Ricci flow approach.
result Existence and uniqueness theorem for solutions to Ricci flow on weighted graphs.
Riemannian flows model curved spaces better.
problem Normalizing flows misspecified on curved spaces.
method Riemannian continuous normalizing flows using ODE solutions.
result Improves modeling on spheres, torii, and hyperbolic spaces.
We consider flows, called Wu flows, whose orbits are the unstable manifolds of a codimension one Anosov flow. Under some regularity assumptions, we give a short proof of the strong mixing property of Wu flows and we show that Wu flows have purely absolutely continuous spectrum in the orthocom…
Proves convergence of mean curvature flow on cylinders with unique continuation.
problem Understanding the convergence and uniqueness of mean curvature flow on cylindrical surfaces.
method Proves convergence and provides unique continuation results for mean curvature flow on cylinders.
result Proves that rescaled mean curvature flow on cylinders converging super-exponentially must coincide with the cylinder itself.
Continuous-time Kyle model shows privacy subsidy from noise-perturbed order flow.
problem Quantifying break-even fees for committed-AMM exchanges under privacy-aggregated information.
method Extended Nakamura's (2026) single-period result to continuous-time, observing order flow perturbed by Brownian noise.
result Cumulative privacy subsidy is identified as equivalent to Loss-Versus-Rebalancing in price observation gap.
Study shows how neck pinches occur in Lagrangian flows and their continuation.
problem Understanding and continuation of Lagrangian mean curvature flows with singularities.
method Analyzes zero Maslov, rational Lagrangian flows in compact Calabi-Yau surfaces.
result Tangent flow is unique and can be continued past singularities.
Under mean curvature flow, a closed, embedded hypersurface M(t) becomes singular in finite time. For certain classes of mean-convex mean curvature flows, we show the continuity of the first singular time T and the limit set "M(T)", with respect to initial data. We employ an Angenent-like neck-pinching argument to…
A new framework solves complex optimization problems with continuous worst-case distributions.
problem Optimizing under uncertain distributions with continuous worst-case scenarios.
method Flow-based distributionally robust optimization (DRO) with Wasserstein uncertainty sets and invertible transport maps.
result The framework finds continuous worst-case distributions and samples efficiently.
The principle of convergence stability for geometric flows is the combination of the continuous dependence of the flow on initial conditions, with the stability of fixed points. It implies that if the flow from an initial state g0 exists for all time and converges to a stable fixed point, then the flows of solutions…
By the method of discrete Morse flows, we construct an energy reducing multiple-valued function flow. The flow we get is Holder continuous with respect to the L-2 norm. We also give another way of constructing flows in some special cases, where the flow we get behaves like ordinary heat flow.
This paper uses normalizing flows to approximate transport maps between densities.
problem Approximating transport maps between given densities.
method Construct time-dependent controls using normalizing flows.
result Provides bounds on the number of switches for piecewise constant approximations.
We consider a continuous path of bounded symmetric Fredholm bilinear forms with arbitrary endpoints on a real Hilbert space, and we prove a formula that gives the spectral flow of the path in terms of the spectral flow of the restriction to a finite codimensional closed subspace. We also discuss the case of restriction…
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.
Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces proved.
problem Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces.
method Analyzing bounded solutions on reduced, locally irreducible compact Kähler spaces.
result Proves continuity of solutions, affirming conjectures and solving open problems.
We consider a continuous curve of self-adjoint Fredholm extensions of a curve of closed symmetric operators with fixed minimal domain Dm and fixed {\it intermediate} domain DW. Our main example is a family of symmetric generalized operators of Dirac type on a compact manifold with boundary with varying well-posed…