arXiv research

The study examines the limitations of bi-Lipschitz Normalizing Flows in approximating certain distributions.

Bi-Lipschitz flows approximate a wide range of distributions.

ELF simplifies normalizing flows, making them more efficient and universal.

The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of the mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm becomes smooth instantly along the mean curvature flow. This generalizes the regular…

Sharp Lipschitz bounds for flow-matching and diffusion models with optimal sampling rates.

Lipschitz regularity proved for harmonic map heat flows into CAT(0) spaces.

A generalization of the Flow-box Theorem is given. The assumption of continuous differentiability of the vector field is relaxed to a local Lipschitz condition. The theorem holds in any Banach space.

New method models fat-tailed distributions with anisotropic tail-adaptive flows.

Minimal existence time for Willmore flow established for smooth and weak Lipschitz initial data.

Residual flows are shown to approximate MMD well.

Develops risk measures on Lipschitz spaces for financial positions.

We show the existence of a global unique and analytic solution for the mean curvature flow, the surface diffusion flow and the Willmore flow of entire graphs for Lipschitz initial data with small Lipschitz norm. We also show the existence of a global unique and analytic solution to the Ricci-DeTurck flow on euclidean s…

We show that the isoperimetric profile $h_{g(t)}(ξ)$ 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, $h^2_{g(t)}(ξ)$ is uniform Lipschitz continuous and hence $h_{g(t)}(…

i-DenseNets improve parameter efficiency and performance in density estimation.

Invertible DenseNets improve model efficiency and performance.

Study Ricci-Deturck flow from rough metrics, proving short-time existence.

Study on Gaussian interpolation flows for generative modeling.

Proves rigidity of maps between manifolds with scalar curvature constraints.

Kim-Milman flow map stable under regular target measures

We show that stochastic interpolation flow maps are Lipschitz with a sharp constant.

Flow Matching improves Wasserstein 1 distance convergence in high dimensions.

We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in $\R^{2n}$, we show that the parabolic equation \eqref{PMA} for the Lagrangian potential has a longtime solution which is sm…

Study shows limitations of Lie bracket commutation for nonsmooth vector fields.

New method proves heat flow of harmonic maps into CAT(0) spaces.

New Transformers maintain Lipschitz continuity for robustness.

Generative algorithms learn high-dimensional data efficiently and generate new samples.

The asymptotic Plateau problem asks for the existence of smooth complete hypersurfaces of constant mean curvature with prescribed asymptotic boundary at infinity in the hyperbolic space $\mathbb{H}^{n+1}$. The modified mean curvature flow (MMCF) was firstly introduced by Xiao and the second author a few years back, and…

The paper analyzes neural network dynamics after weights escape the origin.

The curse of dimensionality affects neural network optimization, especially with smooth functions.

The paper proves smoothness of transition layers in the Allen-Cahn equation.

Normalizing flows optimize Jacobian determinant for unique likelihood objective.

This paper analyzes convergence of large-scale Transformers with weight decay.

Flow-based generative models parameterize probability distributions through an invertible transformation and can be trained by maximum likelihood. Invertible residual networks provide a flexible family of transformations where only Lipschitz conditions rather than strict architectural constraints are needed for enforci…

We investigate the ability of popular flow based methods to capture tail-properties of a target density by studying the increasing triangular maps used in these flow methods acting on a tractable source density. We show that the density quantile functions of the source and target density provide a precise characterizat…

New methods improve efficiency of sampling algorithms for complex systems.

Abstract cone operators prove scalar curvature comparisons on singular manifolds.

Study geometric flows with varying parameters and prove continuous dependence.

We prove the existence of the flow by curvature of regular planar networks starting from an initial network which is non-regular. The proof relies on a monotonicity formula for expanding solutions and a local regularity result for the network flow in the spirit of B. White's local regularity theorem for mean curvature …

In this paper, we prove that if the geodesic flow of a complete manifold without conjugate points with sectional curvatures bounded below by $-c^2$ is of Anosov type, then the constant of contraction of the flow is $\geq e^{-c}$. Moreover, if $M$ has finite volume, the equality holds if and only if the sectional curvat…

We investigate the existence of weak expanding solutions of the harmonic map flow for maps with values into a smooth closed Riemannian manifold. We prove the existence of such solutions in case the target manifold is isometrically embedded as a hypersurface of some Euclidean space and the initial condition is a Lipschi…

Let $M$ be a complete Riemannian manifold which either is compact or has a pole, and let $\varphi$ be a positive smooth function on $M$. In the warped product $M\times_\varphi\mathbb R$, we study the flow by the mean curvature of a locally Lipschitz continuous graph on $M$ and prove that the flow exists for all time an…

We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the $L^2$-space produces the same evolution as the gradient flow of the relative entropy in the $L^2$-Wasserstein space. This means that the heat flow is well defined by either one of the two gradient fl…

Gradient flow of curve length on Sobolev metrics preserves convexity.

We give a new proof of Brakke's partial regularity theorem up to C^{1,ς} for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and an…

We model how Lipschitz continuity changes during neural network training.

Constructs retractions of CAT(1) spaces to convex subsets.

Analyzes Willmore flow for graphs with boundary data, proving existence and convergence.

Generative models improve for multiscale scientific data with new noise and interpolation techniques.