arXiv research

IsUMap improves data visualization of complex geometries.

LDLE embeds manifolds in lower dimensions with low distortion.

A framework compares image representations based on local geometry.

Researchers found 5 local fields to uniquely describe 3D director fields, related through 6 differential relations.

Let X be a path-connected topological space admitting a universal cover. Let Homeo(X,a) denote the group of homeomorphisms of X preserving degree one cohomology class a. We investigate the distortion in Homeo(X,a). Let g be an element of Homeo(X,a). We define a Nielsen-type equivalence relation on the space of g-invari…

We propose a definition of quasi-local mass based on the Penrose Inequality. Two further definitions are given by measuring distortions of the exponential map.

We introduce a new notion of conditional nonlinear expectation under probability distortion. Such a distorted nonlinear expectation is not sub-additive in general, so it is beyond the scope of Peng's framework of nonlinear expectations. A more fundamental problem when extending the distorted expectation to a dynamic se…

New adversarial examples with structured distortion sets improve robustness and perceptibility.

Study on risk measures using distorted Choquet integrals with random distortions.

Paper introduces new risk measures that unify two existing types.

New framework uses cohomology to analyze probabilistic distortions and arbitrage.

We study the problem asking if one can embed manifolds into finite dimensional Euclidean spaces by taking finite number of eigenvector fields of the connection Laplacian. This problem is essential for the dimension reduction problem in massive data analysis. Singer-Wu proposed the vector diffusion map which embeds mani…

QC-SPHARM detects Alzheimer's Disease early using hippocampal surface geometry.

It is known that the surface of a cone over the unit disc with large height has smaller distortion than the standard embedding of the 2-sphere in $\mathbb R^3$. In this note we show that distortion minimisers exist among convex embedded 2-spheres and have uniformly bounded eccentricity. Moreover, we prove that $π/2$ is…

We study the statistical meaning of the minimization of distortion measure and the relation between the equilibrium points of the SOM algorithm and the minima of distortion measure. If we assume that the observations and the map lie in an compact Euclidean space, we prove the strong consistency of the map which almost …

Discover equations of motion from distorted video frames.

LP-FT improves personalized model training in FL by balancing generalization and personalization.

HLoOP detects outliers in hyperbolic 2-space.

New metrics derived from Hölder distortion on Hitchin components.

Curvature regularization prevents distortion in graph embeddings.

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

MLDL preserves manifold geometry in vector transformations.

New method estimates rate-distortion function using optimal transport.

We propose the first global accelerated gradient method for Riemannian manifolds. Toward establishing our result we revisit Nesterov's estimate sequence technique and develop an alternative analysis for it that may also be of independent interest. Then, we extend this analysis to the Riemannian setting, localizing the …

Researchers prove inner product recovery is impossible in latent space models.

New bounds on generalization error for distributed learning using rate-distortion theory.

Given a space $Y$ in $X$, a cycle in $Y$ may be filled with a chain in two ways: either by restricting the chain to $Y$ or by allowing it to be anywhere in $X$. When the pair $(G,H)$ acts on $(X, Y)$, we define the $k$-volume distortion function of $H$ in $G$ to measure the large-scale difference between the volumes of…

Maps and embeddings between hyperbolic spaces and their boundaries studied.

Simple regional perturbations maintain model transferability while reducing adversarial example distortion.

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…

Optimizes maps with controlled distortion for geometric tasks.

Geometric framework analyzes bias in variational inference for posterior functionals.

Study on costs of manipulating AMM-based price oracles.

Neural Local Wasserstein Regression models distribution-on-distribution regression with flexible, localized transport maps.

Vertex distortion detects if a knot is unknot.

What length of rope (of given diameter) is required to tie a particular knot? To answer this question, we define some new notions of thickness for a space curve, one based on Gromov's distortion, and another generalizing the thickness of Litherland, Simon et al. We prove a basic inequality between these thickness measu…

We propose computationally efficient encoders and decoders for lossy compression using a Sparse Regression Code. The codebook is defined by a design matrix and codewords are structured linear combinations of columns of this matrix. The proposed encoding algorithm sequentially chooses columns of the design matrix to suc…

Maps persistence diagrams into Hilbert and Euclidean spaces with explicit distortions.

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…

In this note we obtain local derivative estimates of Shi-type for the heat equation coupled to the Ricci flow. As applications, in part combining with Kuang's work, we extend some results of Zhang and Bamler-Zhang including distance distortion estimates and a backward pseudolocality theorem for Ricci flow on compact ma…

Foundation models fail to preserve continuous geometry, identified as the Geometric Alignment Tax.

We produce examples of codimension one foliations of the Euclidean and hyperbolic planes with bounded geometry which are topologically products, but for which leaves are non-recursively distorted. That is, the function which compares intrinsic distances in leaves with extrinsic distances in the ambient space grows fast…

Ricci flow smooths locally collapsing manifolds with controlled curvature.

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 …

For a connected locally path-connected topological space $X$ and a continuous function $f$ on it such that its Reeb graph $R_f$ is a finite topological graph, we show that the cycle rank of $R_f$, i.e., the first Betti number $b_1(R_f)$, in computational geometry called \emph{number of loops}, is bounded from above by …

It is well-known that quasi-isometries between R-trees induce power quasi-symmetric homeomorphisms between their ultrametric end spaces. This paper investigates power quasi-symmetric homeomorphisms between bounded, complete, uniformly perfect, ultrametric spaces (i.e., those ultrametric spaces arising up to similarity …

Algorithm finds optimal affine transformation to minimize overall distortion.

The Fisher information matrix (FIM) is a fundamental quantity to represent the characteristics of a stochastic model, including deep neural networks (DNNs). The present study reveals novel statistics of FIM that are universal among a wide class of DNNs. To this end, we use random weights and large width limits, which e…