arXiv research

Unsupervised method discovers interpretable directions in GAN latent space.

In this paper, we characterize and classify all surfaces endowed with canonical principal direction relative to a space-like and light-like, constant direction in Minkowski 3-spaces.

Alexandrov spaces have a special stratification that maps to spheres.

We study direct limits $(G,K) = \varinjlim (G_n,K_n)$ of compact Gelfand pairs. First, we develop a criterion for a direct limit representation to be a multiplicity--free discrete direct sum of irreducible representations. Then we look at direct limits $G/K = \varinjlim G_n/K_n$ of compact riemannian symmetric spaces, …

The study characterizes and analyzes spacelike surfaces with a canonical normal null direction in Minkowski 4-space.

The study examines principal directions and curvatures of Lagrangian submanifolds.

Estimates modes and ridges in mixed Euclidean and directional spaces.

In this paper, we introduce canonical principal direction (CPD) submanifolds with higher codimension in Euclidean spaces. We obtain the complete classification of surfaces endowed with CPD in the Euclidean 4-space.

New Teichmüller geodesic rays found with unique foliations.

In the bordered Floer theory, gluing thickened torus of positive meridional Dehn twist to the boundary of a knot complement result in the knot complement of increased framing. For a fixed knot K, we construct a direct system of positively framed knot complements and study the direct limit. We also study the morphism sp…

We analyze the information-theoretic limits for the recovery of node labels in several network models. This includes the Stochastic Block Model, the Exponential Random Graph Model, the Latent Space Model, the Directed Preferential Attachment Model, and the Directed Small-world Model. For the Stochastic Block Model, the…

In this study, we define a new type of direction curves in the Euclidean 3-space such as osculating-direction curve. We give the characterizations for these curves. Moreover, we obtain the relationships between osculating direction curves and some special curves such as helix, slant helix or rectifying curves.

Geometric quantization often produces not one Hilbert space to represent the quantum states of a classical system but a whole family $H_s$ of Hilbert spaces, and the question arises if the spaces $H_s$ are canonically isomorphic. [ADW] and [Hi] suggest to view $H_s$ as fibers of a Hilbert bundle $H$, introduce a connec…

The paper studies how spaces collapse to Alexandrov spaces with mild singularities.

A particular Finsler-metric proposed in [1,2] and describing a geometry with a preferred null direction is characterized here as belonging to a subclass contained in a larger class of Finsler-metrics with one or more preferred directions (null, space- or timelike). The metrics are classified according to their group of…

Enhances interpretability of linear latent spaces through automated clustering and ranking.

We show that the cobordism groups of negative codimensional folds maps contain direct sums of stable homotopy groups of Thom spaces of vector bundles like the circle and the infinite dimensional projective space. We give geometrical invariants which detect these direct summands.

Introduces Alexandrov spaces with curvature below, covering various theorems.

Given a constant vector field $Z$ in Minkowski space, a timelike surface is said to have a canonical null direction with respect to $Z$ if the projection of $Z$ on the tangent space of the surface gives a lightlike vector field. In this paper we describe these surfaces in the ruled case. For example when the Minkowski …

The paper presents a new method to represent directed graphs using pseudo-Riemannian manifolds.

A framework visualizes embedding spaces of neural survival analysis models using anchor directions.

LayerNorm transformers have dead directions that can be read from their parameters alone.

Study embeddings of free group products into automorphism groups.

Study classifies translating solitons in Minkowski 3-space, revealing singularities.

We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure …

Given a vector field $X$ in a Riemannian manifold, a hypersurface is said to have a canonical principal direction relative to $X$ if the projection of $X$ onto the tangent space of the hypersurface gives a principal direction. We give different ways for building these hypersurfaces, as well as a number of useful charac…

In the present paper we classify all surfaces in $\E^3$ with a canonical principal direction. Examples of these type of surfaces are constructed. We prove that the only minimal surface with a canonical principal direction in the Euclidean space ${\mathbb{E}}^3$ is the catenoid.

We highlight recent progresses in the study of the Weil-Petersson (WP) geometry of finite dimensional Teichmüller spaces. For recent progress on and the understanding of infinite dimensional Teichmüller spaces the reader is directed to the recent work of Teo-Takhtajan. As part of the highlight, we also present possible…

Study on Kuranishi spaces of complex structures and vector bundles, showing isomorphisms and counterexamples.

New example disproves complex contact theory for fat distributions with Reeb directions.

We show that directed minimal cones in (n+1)-dimensional Euclidean space which have at most one singularity are - besides the trivial cases: empty set, whole space - half spaces. Using blow-up techniques, this result can be used to get C^{1,lambda}-regularity for the measure-theoretic boundary of almost minimal Cacciop…

Extends Moutard quadric concept to higher dimensions.

Paper studies metric ribbon graphs and provides a recursion for their volumes.

A directed curve is a possibly singular curve with well-defined tangent lines along the curve. Then the tangent surface to a directed curve is naturally defined as the ruled surface by tangent geodesics to the curve, whenever any affine connection is endowed with the ambient space. In this paper the local diffeomorphis…

ConMeZO speeds up zeroth-order optimization for large language models.

Gradient-free method solves infinite-dimensional optimization problems.

This paper considers the problem of embedding directed graphs in Euclidean space while retaining directional information. We model a directed graph as a finite set of observations from a diffusion on a manifold endowed with a vector field. This is the first generative model of its kind for directed graphs. We introduce…

FastMap-D embeds directed graphs using potential fields.

Let $S$ be a complete flat surface, such as the Euclidean plane. We obtain direct characterizations of the connected components of the space of all curves on $S$ which start and end at given points in given directions, and whose curvatures are constrained to lie in a given interval, in terms of all parameters involved.…

We study direct limits $(G,K) = \varinjlim (G_n,K_n)$ of Gelfand pairs of the form $G_n = N_n\rtimes K_n$ with $N_n$ nilpotent, in other words pairs $(G_n,K_n)$ for which $G_n/K_n$ is a commutative nilmanifold. First, we extend the criterion of \cite{W3} for a direct limit representation to be multiplicity free. Then w…

Paper introduces AIF for anomaly detection with variable feature sensitivity.

In this note we give a construction of a smooth Riemannian metric on R^n which is standard Euclidean outside a compact set K and such that it has N = n(n + 1)=2 invisible directions, meaning that all geodesics lines passing through the set K in these directions remain the same straight lines on exit. For example in the…

We present a construction that produces infinite classes of Kähler groups that arise as fundamental groups of fibres of maps to higher dimensional tori. Following the work of Delzant and Gromov, there is great interest in knowing which subgroups of direct products of surface groups are Kähler. We apply our construction…

The paper studies circular evolutes and involutes of framed curves in Euclidean space.

CDFD analyzes circularity and directionality in weighted directed networks.

Develops geometric foundations for sublinear Morse boundaries in mapping class groups and Teichmüller spaces.

Distributions over permutations arise in applications ranging from multi-object tracking to ranking of instances. The difficulty of dealing with these distributions is caused by the size of their domain, which is factorial in the number of considered entities ($n!$). It makes the direct definition of a multinomial dist…

We develop some results on the positivity of direct image bundles in the particular case of a trivial fibration over a one-dimensional base. We also apply the results to study variations of Kahler metrics.