arXiv research

Inversive distance circle packing metric was introduced by P Bowers and K Stephenson \cite{BS} as a generalization of Thurston's circle packing metric \cite{T1}. They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo \cite{Guo} proved the infinitesimal rigidity a…

Proves rigidity of circle packings in the plane, generalizing previous work.

We give a counterexample of Bowers-Stephenson's conjecture in the spherical case: spherical inversive distance circle packings are not determined by their inversive distances.

Paper proves circle packings converge to Riemann mapping for Jordan domains.

Inversive distance circle packing on surfaces was introduced by Bowers-Stephenson as a generalization of Thurston's circle packing and conjectured to be rigid. The infinitesimal and global rigidity of circle packing with nonnegative inversive distance were proved by Guo and Luo respectively. The author proved the globa…

This work characterizes, analytically and numerically, two major effects of the quadratic Wasserstein ($W_2$) distance as the measure of data discrepancy in computational solutions of inverse problems. First, we show, in the infinite-dimensional setup, that the $W_2$ distance has a smoothing effect on the inversion pro…

The paper introduces a new Wasserstein distance for approximating posteriors in inverse problems.

This paper investigates several global rigidity issues for polyhedral surfaces including inversive distance circle packings. Inversive distance circle packings are polyhedral surfaces introduced by P. Bowers and K. Stephenson as a generalization of Andreev-Thurston's circle packing. They conjectured that inversive dist…

A Euclidean (or hyperbolic) circle packing on a closed triangulated surface with prescribed inversive distance is locally determined by its cone angles. We prove this by applying a variational principle.

In this paper, we generalize Chow-Luo's combinatorial Ricci flow to inversive distance circle packing setting. Although the solution to the generalized flow may develop singularities in finite time, we can always extend the solution so as it exists for all time and converges exponentially fast. Thus the generalized flo…

The paper proves the existence of a unique circle packing on hyperbolic surfaces.

The subject of this article is the introduction of a new concept of well-posedness of Bayesian inverse problems. The conventional concept of (Lipschitz, Hellinger) well-posedness in [Stuart 2010, Acta Numerica 19, pp. 451-559] is difficult to verify in practice and may be inappropriate in some contexts. Our concept sim…

Given a triangulated surface $M$, we use Ge-Xu's $α$-flow \cite{Ge-Xu1} to deform any initial inversive distance circle packing metric to a metric with constant $α$-curvature. More precisely, we prove that the inversive distance circle packing with constant $α$-curvature is unique if $αχ(M)\leq 0$, which generalize And…

The aim of the paper is to investigate the relation between inverse limit of branched manifolds and codimension zero laminations. We give necessary and sufficient conditions for such an inverse limit to be a lamination. We also show that codimension zero laminations are inverse limits of branched manifolds. The inverse…

The paper extends the Discrete Schwarz-Pick Lemma to circle packings with obtuse intersections and disjoint packings.

The MEM method uses data-driven priors for linear inverse problems, proving convergence and estimating differences.

Reconstructing manifolds from partial distance and heat kernel data.

IDA adapts to non-iid data in federated learning for medical imaging.

Generative models improve inverse problems by providing tailored priors.

Graph curvature measured by inverse resistance distance.

Unified framework recovers exact input from SOM activation patterns.

New theorem proves rigidity of circle packings in hyperbolic geometry.

Study improves estimation of functions from noisy data using convex penalties.

Global optimization problems whose objective function is expensive to evaluate can be solved effectively by recursively fitting a surrogate function to function samples and minimizing an acquisition function to generate new samples. The acquisition step trades off between seeking for a new optimization vector where the…

New method finds metrics on surfaces with prescribed curvatures using circle packings and surgery.

We present constructions inspired by the Ma-Schlenker example of~\cite{Ma:2012hl} that show the non-rigidity of spherical inversive distance circle packings. In contrast to the use in~\cite{Ma:2012hl} of an infinitesimally flexible Euclidean polyhedron, embeddings in de Sitter space, and Pogorelov maps, our elementary …

Study shows stability of travel time data reconstruction from closed subsets.

This paper reviews SDR methods for multivariate response regression.

Hierarchical clustering uses OWA operators to generalize linkage methods and avoid dendrogram inversions.

Let $(N,g)$ be a Riemannian manifold with the distance function $d(x,y)$ and an open subset $M\subset N$. For $x\in M$ we denote by $D_x$ the distance difference function $D_x:F\times F\to \mathbb R$, given by $D_x(z_1,z_2)=d(x,z_1)-d(x,z_2)$, $z_1,z_2\in F=N\setminus M$. We consider the inverse problem of determining …

Proves existence of unique circle packings on polyhedral surfaces.

The paper shows how to use hyperplanes and hyperballs interchangeably using inversive geometry.

We consider an enlarged dimension reduction space in functional inverse regression. Our operator and functional analysis based approach facilitates a compact and rigorous formulation of the functional inverse regression problem. It also enables us to expand the possible space where the dimension reduction functions bel…

Proof of existence and uniqueness of weighted Voronoi-Delaunay on polyhedral surfaces.

A method detects vehicles far from tunnel CCTV using AI.

We study surfaces with decorations and prove uniformization in non-Euclidean geometries.

The problem of detecting data anomaly is considered. Under the null hypothesis that models anomaly-free data, measurements are assumed to be from an unknown distribution with some authenticated historical samples. Under the composite alternative hypothesis, measurements are from an unknown distribution positive distanc…

This paper presents a general notion of Mahalanobis distance for functional data that extends the classical multivariate concept to situations where the observed data are points belonging to curves generated by a stochastic process. More precisely, a new semi-distance for functional observations that generalize the usu…

Researchers reconstruct simple Riemannian manifolds from boundary wave arrival times.

In this paper, we discuss the uniqueness in an integral geometry problem in a strongly convex domain. Our problem is related to the problem of finding a Riemannian metric by the distances between all pairs of the boundary points. For the proof, the problem is reduced to an inverse source problem for a kinetic equation …

A new framework enhances IDW models for complex industrial datasets.

We consider various notions of strains; quantitative measures for the deviation of a linear transformation from an isometry. The main approach, which is motivated by physical applications and follows the work of Patrizio Neff and co-workers , is to select a Riemannian metric on $\text{GL}_n$, and use its induced geodes…

Unified framework for lifted training and inversion of neural networks.

The paper analyzes Tikhonov regularization in Hilbert scales for statistical inverse problems.

Bi-geodesic mappings preserve distances on hyperbolic surfaces with boundaries.

Fundamental weight systems identified as quantum states.

In this paper, we introduce a new combinatorial curvature on triangulated surfaces with inversive distance circle packing metrics. Then we prove that this combinatorial curvature has global rigidity. To study the Yamabe problem of the new curvature, we introduce a combinatorial Ricci flow, along which the curvature evo…

Paper tackles estimating initial conditions of spatio-temporal processes from sparse data.