arXiv research

The paper extends a link criterion for Lipschitz normal embeddings to definable sets in o-minimal structures.

The study examines Lipschitz normally embedded Hölder triangles in 4D space.

Analytic sets with unique infinite tangent cone are algebraic.

Study conic singular manifolds, proving Lipschitz normal embedding.

Conic singular sub-manifolds are Lipschitz Normally Embedded in compact non-Euclidean manifolds.

Characterizes hypergenerated stratified groups with flat boundaries.

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

Defines smoothness of definable sets in o-minimal structures.

Characterizes quasi-isometric embeddings in coarsely Lipschitz category.

Bi-Lipschitz flows approximate a wide range of distributions.

Lipschitz continuity recently becomes popular in generative adversarial networks (GANs). It was observed that the Lipschitz regularized discriminator leads to improved training stability and sample quality. The mainstream implementations of Lipschitz continuity include gradient penalty and spectral normalization. In th…

Paper proves convergence for private FL on non-Lipschitz convex objectives using normalization instead of clipping.

Let $X$ be a closed semialgebraic set of dimension $k.$ If $n\ge 2k+1$, then there is a bi-Lipschitz and semialgebraic embedding of $X$ into $\Bbb R^n.$ Moreover, if $n \ge 2k+2$, then this embedding is unique (up to a bi-Lipschitz and semialgebraic homeomorphism of $\Bbb R^n.$

This paper examines weight initialization for 1-Lipschitz networks to improve robustness against adversarial attacks.

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)}(…

Lipschitz normalization boosts deep attention models, especially for graph neural networks.

GraN-GAN normalizes gradients for better GAN performance.

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

The paper extends Lipschitz metric isometries between Outer Spaces of virtually free groups.

We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash $C^1$ Embedding Theorem. For more general metric spaces the same result is false, e.g., for Finsler non-Riemannian manifolds. However, we also sh…

Proves isometric embeddings in Euclidean spaces for RCD spaces.

The study examines the normal growth exponent of submanifolds in negatively curved manifolds.

Normalization layers control deep neural network capacity, improving stability and generalization.

Residual flows are shown to approximate MMD well.

The paper proves the existence of hypersurfaces with prescribed mean curvature.

We construct bi-Lipschitz embeddings into Euclidean space for manifolds and orbifolds of bounded diameter and curvature. The distortion and dimension of such embeddings is bounded by diameter, curvature and dimension alone. Our results also apply for bounded subsets of complete Riemannian manifolds, and complete flat a…

Nilpotent groups can't be biLipschitz embedded into $L^1$.

The paper extends quasimorphisms on subgroups to larger groups.

SRNF framework extends surface distance to Lipschitz surfaces.

For all $k,n\ge 1$, we construct a biLipschitz embedding of $\mathbb{S}^n$ into the jet space Carnot group $J^k(\mathbb{R}^n)$ that does not admit a Lipschitz extension to $\mathbb{B}^{n+1}$. Let $f:\mathbb{B}^n\to \mathbb{R}$ be a smooth, positive function with $k^{th}$-order derivatives that are approximately linear …

Develops a new framework for temporal anchoring in deep embedding spaces.

Generative adversarial networks (GANs) are one of the most popular approaches when it comes to training generative models, among which variants of Wasserstein GANs are considered superior to the standard GAN formulation in terms of learning stability and sample quality. However, Wasserstein GANs require the critic to b…

We provide a characterization of r-regular sets in terms of the Lipschitz regularity of normal vector fields to the boundary.

Continuous MDS embeds sequences of dissimilarities in Euclidean space.

Study improves Poincaré-Sobolev inequalities for differential forms.

Exciting new work on the generalization bounds for neural networks (NN) given by Neyshabur et al. , Bartlett et al. closely depend on two parameter-depenedent quantities: the Lipschitz constant upper-bound and the stable rank (a softer version of the rank operator). This leads to an interesting question of whether cont…

Curvature measures uniquely determined by invariance under embeddings.

We show that a family of isolated complex hypersurface singularities with constant Milnor number may fail, in the strongest sense, to have constant bi-Lipschitz type. Our example is the Briac con--Speder family $X_t:=\{(x,y,z)\in\C^3 | x^5+z^{15}+y^7z+txy^6=0 \}$ of normal complex surface germs; we show the germ $(X_0,…

Hyperbolic space outperforms Euclidean in learning hierarchical data.

This paper presents a margin-based multiclass generalization bound for neural networks that scales with their margin-normalized "spectral complexity": their Lipschitz constant, meaning the product of the spectral norms of the weight matrices, times a certain correction factor. This bound is empirically investigated for…

We discuss a variation of Gromov's notion of asymptotic dimension that was introduced and named Nagata dimension by Assouad. The Nagata dimension turns out to be a quasisymmetry invariant of metric spaces. The class of metric spaces with finite Nagata dimension includes in particular all doubling spaces, metric trees, …

We show that, for all $α\geq 0$, the generalized Grushin plane $\mathbb{G}_α$ is bi-Lipschitz homeomorphic to a $2$-dimensional quasiplane in the Euclidean space $\mathbb{R}^{[α]+2}$, where $[α]$ is the integer part of $α$. The target dimension is sharp. This generalizes a recent result of Wu.

In the Engel group with its Carnot group structure we study subsets of locally finite subRiemannian perimeter and possessing constant subRiemannian normal. We prove the rectifiability of such sets: more precisely we show that, in some specific coordinates, they are upper-graphs of entire Lipschitz functions (with respe…

The Nash-Kuiper Theorem states that the collection of $C^1$-isometric embeddings from a Riemannian manifold $M^n$ into $\mathbb{E}^N$ is $C^0$-dense within the collection of all smooth 1-Lipschitz embeddings provided that $n < N$. This result is now known to be a consequence of Gromov's more general $h$-principle. Ther…

Normalizing flows optimize Jacobian determinant for unique likelihood objective.

Study shows shallow ReLU networks struggle with high-dimensional Lipschitz functions.

For a normal covering over a closed oriented topological manifold we give a proof of the L2-signature theorem with twisted coefficients, using Lipschitz structures and the Lipschitz signature operator introduced by Teleman. We also prove that the L-theory isomorphism conjecture as well as the C^*_max-version of the Bau…

Study on Gaussian interpolation flows for generative modeling.