arXiv research

The paper extends circle pattern flows to hyperbolic and Euclidean geometry.

Defines surfaces with specific contour patterns.

A simple method makes Euclidean patterns look like Escher's art.

In this paper we give two different proofs of Bobenko and Springborn's theorem of circle pattern: there exists a hyperbolic (or Euclidean) circle pattern with proscribed intersection angles and cone angles on a cellular decomposed surface up to isometry (or similarity).

Study of combinatorial Calabi flow on ideal circle patterns.

The understanding of geographical reality is a process of data representation and pattern discovery. Former studies mainly adopted continuous-field models to represent spatial variables and to investigate the underlying spatial continuity/heterogeneity in the regular spatial domain. In this article, we introduce a more…

We consider ``hyperideal'' circle patterns, i.e. patterns of disks appearing in the definition of the Delaunay decomposition associated to a set of disjoint disks, possibly with cone singularities at the center of those disks. Hyperideal circle patterns are associated to hyperideal hyperbolic polyhedra. We describe the…

The edge of torn elastic sheets and growing leaves often form a hierarchical buckling pattern. Within non-Euclidean plate theory this complex morphology can be understood as low bending energy isometric immersions of hyperbolic Riemannian metrics. With this motivation we study the isometric immersion problem in strip a…

This work proposes hyperbolic deep convolutional neural networks for better pattern recognition.

Unique circle patterns on spheres found for spherical conical metrics.

We prove existence and uniqueness results for patterns of circles with prescribed intersection angles in constant curvature surfaces. Our method is based on two new functionals--one for the Euclidean and one for the hyperbolic case. We show how Colin de Verdi`ere's, Br"agger's and Rivin's functionals can be derived fro…

Driving styles have a great influence on vehicle fuel economy, active safety, and drivability. To recognize driving styles of path-tracking behaviors for different divers, a statistical pattern-recognition method is developed to deal with the uncertainty of driving styles or characteristics based on probability density…

New method finds ideal circle patterns on spheres.

Paper uses non-Euclidean analysis to classify brain structure variations.

Study infinite circle patterns in the Weil-Petersson class using discrete harmonic functions.

The edges of torn plastic sheets and growing leaves often display hierarchical buckling patterns. We show that this complex morphology (i) emerges even in zero strain configurations, and (ii) is driven by a competition between the two principal curvatures, rather than between bending and stretching. We identify the key…

We consider the empirical risk minimization problem for linear supervised learning, with regularization by structured sparsity-inducing norms. These are defined as sums of Euclidean norms on certain subsets of variables, extending the usual $\ell_1$-norm and the group $\ell_1$-norm by allowing the subsets to overlap. T…

Consider a curve $Γ$ in a domain $D$ in the plane $\boldsymbol R^2$. Thinking of $D$ as a piece of paper, one can make a curved folding $P$ in the Euclidean space $\boldsymbol R^3$. The singular set $C$ of $P$ as a space curve is called the crease of $P$ and the initially given plane curve $Γ$ is called the crease patt…

Survey on discrete minimal surfaces and their properties.

New discrete cmc surfaces defined from sphere packings and combinatorics.

Paper resolves spherical curvature flow problem.

Novel defects in hyperbolic sheets explain complex wrinkling patterns in nature.

A new associative memory uses Sinkhorn divergence for efficient pattern retrieval.

This work improves KG embeddings by integrating hyperbolic and attention mechanisms.

Deep neural networks show some layers better align with data than others.

A Delaunay cell decomposition of a surface with constant curvature gives rise to a circle pattern, consisting of the circles which are circumscribed to the facets. We treat the problem whether there exists a Delaunay cell decomposition for a given (topological) cell decomposition and given intersection angles of the ci…

A hybrid K-NN and SVM technique improves classification accuracy.

This paper introduces an approach for detecting differences in the first-order structures of spatial point patterns. The proposed approach leverages the kernel mean embedding in a novel way by introducing its approximate version tailored to spatial point processes. While the original embedding is infinite-dimensional a…

This paper provides a general framework to study the effect of sampling properties of training data on the generalization error of the learned machine learning (ML) models. Specifically, we propose a new spectral analysis of the generalization error, expressed in terms of the power spectra of the sampling pattern and t…

HypeGBMS clusters data in hyperbolic space, overcoming Euclidean limitations.

Discrete Laplacians defined for spherical and hyperbolic surfaces.

Consider an oriented curve $Γ$ in a domain $D$ in the plane $\boldsymbol R^2$. Thinking of $D$ as a piece of paper, one can make a curved folding in the Euclidean space $\boldsymbol R^3$. This can be expressed as the image of an "origami map" $Φ:D\to \boldsymbol R^3$ such that $Γ$ is the singular set of $Φ$, the word "…

CEDA analyzes large categorical datasets using tree geometry and binary codes.

The one-class classification problem is a well-known research endeavor in pattern recognition. The problem is also known under different names, such as outlier and novelty/anomaly detection. The core of the problem consists in modeling and recognizing patterns belonging only to a so-called target class. All other patte…

Discrete conformal maps on surfaces with vertex decorations are studied.

It is often hypothesized that a crucial role for recurrent connections in the brain is to constrain the set of possible response patterns, thereby shaping the neural code. This implies the existence of neural codes that cannot arise solely from feedforward processing. We set out to find such codes in the context of one…

Imaging-based early diagnosis of Alzheimer Disease (AD) has become an effective approach, especially by using nuclear medicine imaging techniques such as Positron Emission Topography (PET). In various literature it has been found that PET images can be better modeled as signals (e.g. uptake of florbetapir) defined on a…

In this paper, we propose a novel adaptive kernel for the radial basis function (RBF) neural networks. The proposed kernel adaptively fuses the Euclidean and cosine distance measures to exploit the reciprocating properties of the two. The proposed framework dynamically adapts the weights of the participating kernels us…

The problem of joint feature selection across a group of related tasks has applications in many areas including biomedical informatics and computer vision. We consider the l2,1-norm regularized regression model for joint feature selection from multiple tasks, which can be derived in the probabilistic framework by assum…

The paper explores how different patterns of heterophily affect Graph Neural Networks.

Research shows that certain metric spaces cannot contain rigid structures and provides evidence for loose embeddings into Euclidean spaces.

Traditional anatomical analyses captured only a fraction of real phenomic information. Here, we apply deep learning to quantify total phenotypic similarity across 2468 butterfly photographs, covering 38 subspecies from the polymorphic mimicry complex of $\textit{Heliconius erato}$ and $\textit{Heliconius melpomene}$. E…

Unified framework recovers exact input from SOM activation patterns.

Graph neural networks improve predictions on graph data.

We found a class of triangulated surfaces in Euclidean space which have similar properties as isothermic surfaces in Differential Geometry. We call a surface isothermic if it admits an infinitesimal isometric deformation preserving the mean curvature integrand locally. We show that this class is Möbius invariant. Isoth…

Consider a 3$-$dimensional manifold $N$ obtained by gluing a finite number of ideal hyperbolic tetrahedra via isometries along their faces. By varying the isometry type of each tetrahedron but keeping fixed the gluing pattern we define a space $\mathcal{T}$ of complete hyperbolic metrics on $N$ with cone singularities …

Curvature regularization prevents distortion in graph embeddings.

Paper improves estimates for discrete Laplace in hyperbolic geometry.