arXiv research

Sharp bounds for distortion risk metrics under uncertain distributions.

New metrics derived from Hölder distortion on Hitchin components.

Revises Gauss's Lemma using metrical distortion and differential slip.

Algorithm finds optimal affine transformation to minimize overall distortion.

Study bounds on curvature for special Finsler metrics.

Small neural networks embed arbitrary metric spaces into Gaussian mixtures.

Study the tradeoff between signal distortion and human perception over finite channels.

Given a metric space $X$ and a function $f: X \to \mathbb{R}$, the Reeb construction gives metric a space $X_f$ together with a quotient map $X \to X_f$. Under suitable conditions $X_f$ becomes a metric graph and can therefore be used as a graph approximation to $X$. The Gromov-Hausdorff distance from $X_f$ to $X$ is b…

The paper analyzes worst-case distortion risk metrics and weighted entropy under partial information.

This paper is a survey of some of the developments in coarse extrinsic geometry since its inception in the work of Gromov. Distortion, as measured by comparing the diameter of balls relative to different metrics, can be regarded as one of the simplist extrinsic notions. Results and examples concerning distorted subgrou…

New method accelerates gradient descent on curved spaces.

The Madry Lab recently hosted a competition designed to test the robustness of their adversarially trained MNIST model. Attacks were constrained to perturb each pixel of the input image by a scaled maximal $L_\infty$ distortion $ε$ = 0.3. This discourages the use of attacks which are not optimized on the $L_\infty$ dis…

In this paper, we define a new metric structure on the shape space of a high genus surface. We introduce a rigorous definition of a shape of a surface and construct a metric based on two energies measuring the area distortion and the angle distortion of a quasiconformal homeomorphism. We show that the energy minimizer …

A new distortion measure optimizes function approximations in vector quantization.

In this note, we prove a uniform distance distortion estimate for Ricci flows with uniformly bounded scalar curvature, independent of the lower bound of the initial $μ$-entropy. Our basic principle tells that once correctly renormalized, the metric-measure quantities obey similar estimates as in the non-collapsing case…

A framework compares image representations based on local geometry.

Hierarchical clustering is a popular method for analyzing data which associates a tree to a dataset. Hartigan consistency has been used extensively as a framework to analyze such clustering algorithms from a statistical point of view. Still, as we show in the paper, a tree which is Hartigan consistent with a given dens…

We exhibit rigid rotations of spheres as distortion elements in groups of diffeomorphisms, thereby answering a question of J Franks and M Handel. We also show that every homeomorphism of a sphere is, in a suitable sense, as distorted as possible in the group Homeo(S^n), thought of as a discrete group. An appendix by Y …

Study shows a mass quantity for $C^0$ metrics that agrees with ADM mass.

IsUMap improves data visualization of complex geometries.

The paper studies quasimorphisms and distortion in homeomorphism groups of manifolds.

Given a finite metric CW complex $X$ and an element $α\in π_n(X)$, what are the properties of a geometrically optimal representative of $α$? We study the optimal volume of $kα$ as a function of $k$. Asymptotically, this function, whose inverse, for reasons of tradition, we call the volume distortion, turns out to be an…

In cognitive psychology, automatic and self-reinforcing irrational thought patterns are known as cognitive distortions. Left unchecked, patients exhibiting these types of thoughts can become stuck in negative feedback loops of unhealthy thinking, leading to inaccurate perceptions of reality commonly associated with anx…

The paper calculates bounds for risk metrics and entropies under partial information constraints.

Linear progress observed in fibered 3-manifold embeddings.

FID misaligns with human judgment due to reliance on ImageNet classes.

New compression theory justifies model pruning for neural networks.

The paper optimizes watermarking of LLMs, improving detection ability.

A morph between two Riemannian $n$-manifolds is an isotopy between them together with the set of all intermediate manifolds equipped with Riemannian metrics. We propose measures of the distortion produced by some classes of morphs and diffeomorphisms between two isotopic Riemannian $n$-manifolds and, with respect to th…

Simplifies VAE for anomaly detection using rate-distortion theory.

Framework uses IRL and RL to elicit and optimize risk preferences robustly to noise.

The paper connects geometric and topological concepts to bound distances between metric spaces.

We analyze Riemannian accelerated methods using a new framework.

Vertex distortion detects if a knot is unknot.

We prove that any diffeomorphism of the sphere S^n to itself can be decomposed into bi-Lipschitz mappings of small isometric distortion and which move points a small amount in the spherical metric.

New method corrects complex distortions in single view images.

Equipped with the L^2-distortion distance, the space "X" of all metric measure spaces (X,d,m) is proven to have nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are characterized in detail. Moreover, classes of semiconvex functionals and their gradient flows on "X" are presented.

Deep generative models provide a systematic way to learn nonlinear data distributions, through a set of latent variables and a nonlinear "generator" function that maps latent points into the input space. The nonlinearity of the generator imply that the latent space gives a distorted view of the input space. Under mild …

Recent studies have highlighted the vulnerability of deep neural networks (DNNs) to adversarial examples - a visually indistinguishable adversarial image can easily be crafted to cause a well-trained model to misclassify. Existing methods for crafting adversarial examples are based on $L_2$ and $L_\infty$ distortion me…

Vertex distortion measures how far lattice knots deviate from straight lines.

LIMP learns latent shapes with metric preservation, improving generative models.

We consider the problem of distortion minimal morphing of $n$-dimensional compact connected oriented smooth manifolds without boundary embedded in $\R^{n+1}$. Distortion involves bending and stretching. In this paper, minimal distortion (with respect to stretching) is defined as the infinitesimal relative change in vol…

This paper shows how to calculate risk measures for sums of two counter-monotonic risks.

We analyze the obstruction to metrics of positive scalar curvature within a given bounded distortion class of metrics. This obstruction lives in a non-Hausdorff cohomology group Poincare dual to the uniformly finite homology studied by Block and Weinberger. One of the applications is a converse to their theorem on infi…

Study distortion risk measures for step-weighted distributions.

We study the Lipschitz metric on Teichmuller space (defined by Thurston) and compare it with the Teichmuller metric. We show that in the thin part of Teichmuller space the Lipschitz metric is approximated up to bounded additive distortion by the sup metric on a product of lower-dimensional spaces (similar to the Teichm…

The distortion of a curve measures the maximum arc/chord length ratio. Gromov showed any closed curve has distortion at least pi/2 and asked about the distortion of knots. Here, we prove that any nontrivial tame knot has distortion at least 5pi/3; examples show that distortion under 7.16 suffices to build a trefoil kno…

Computed distortion coefficients for the α-Grushin plane.