NAM maps images between domains without adversarial learning, achieving better quality and stability.
problem Translating images between domains without correspondences and adversarial learning.
method NAM separates generative modeling and cross-domain mapping, using a pre-trained target model.
result Higher quality and resolution image translations with simpler and more stable training.
Maps preserving Carathéodory distance between symmetric domains are rigid.
problem Rigidity of maps preserving Carathéodory distance between bounded symmetric domains.
method Large-scale geometry of Carathéodory distance, horocompactification, Gromov product.
result Maps preserving Carathéodory distance are rigid and either holomorphic or antiholomorphic.
The paper provides a risk bound for unsupervised cross-domain mapping algorithms.
problem Theoretical support for unsupervised cross-domain mapping algorithms is lacking.
method Adversarial training methods based on IPMs and a novel risk bound.
result A risk bound that upper bounds the risk between learned and target mappings.
Paper introduces a novel map learning algorithm for domain translation and adaptation.
problem Learning a map between related data spaces that can be applied to out-of-sample data and satisfies application-specific constraints.
method Utilizes normalizing flows to parameterize a map that minimizes a probability distance and application-specific regularizers, solving a modified optimal transport problem.
result The proposed method (parOT) outperforms existing optimal transport approaches in domain adaptation and translation tasks.
JSCN improves cross-domain recommendation by learning domain-invariant user representations.
problem Cross-domain recommendation data sparsity and domain-incompatibility issues.
method JSCN uses multi-layer spectral convolutions on different graphs to learn domain-invariant user representations and domain adaptive user mappings.
result Significant improvement in cross-domain recommendation performance (9.2% recall, 36.4% MAP improvements).
Study on harmonic maps from surfaces with energy bounds and neck domains.
problem Behavior of harmonic maps with bounded energy on complex domains.
method Analysis of a sequence of harmonic maps in generalized neck domains.
result Upper bound of energy density and study of nullity and index limits.
Improved method for numerical conformal mappings on complex domains.
problem Accurate and efficient computation of conformal mappings on multiply connected domains.
method Generalization and refinement of the conjugate function method using high-order finite element methods.
result Achieved accurate and efficient construction of boundary values for multiply connected domains.
Paper proves circle packings converge to Riemann mapping for Jordan domains.
problem Proving discrete conformal maps converge to Riemann mapping.
method Establishing solvability theorem for inversive distance circle packings.
result Bowers-Stephenson's conjecture for Jordan domains is proven.
Study proper holomorphic maps between symmetric domains, proving rigidity and isometric properties.
problem Characterizing proper holomorphic maps between symmetric domains.
method Analyzing maps between bounded symmetric domains of the same rank, proving rigidity and isometric properties.
result Proper holomorphic maps are totally geodesic isometric embeddings under certain conditions.
The paper provides a concentration result and sample complexity for linear Monge mapping estimation and its application in domain adaptation.
problem Estimating the linear Monge mapping between distributions and its application in domain adaptation.
method The approach involves proving a concentration result and sample complexity for the linear mapping operator, and using it to derive a generalization bound for domain adaptation with optimal transport.
result The method achieves a sample complexity of n−1/2 and approaches the performance of theoretical Bayes predictor under mild conditions. Two new boundary Schwarz lemmas for holomorphic maps established.
problem Establishing boundary Schwarz lemmas for holomorphic maps.
method Two boundary versions of the Schwarz lemma for holomorphic maps.
result First boundary Schwarz lemmas for general holomorphic self maps and biholomorphisms with invariant Kähler metrics.
We prove local Holder continuity of quasi-n-harmonic mappings from Euclidean domains into metric spaces with non-positive curvature in the sense of Alexandrov. We also obtain global Holder continuity of such mappings from bounded Lipschitz domains.
The Teichmüller harmonic map flow, introduced in [9], evolves both a map from a closed Riemann surface to an arbitrary compact Riemannian manifold, and a constant curvature metric on the domain, in order to reduce its harmonic map energy as quickly as possible. In this paper, we develop the geometric analysis of holomo…
In \cite{kamz} the author proved that every quasiconformal harmonic mapping between two Jordan domains with C1,α, 0<α≤1, boundary is bi-Lipschitz, providing that the domain is convex. In this paper we avoid the restriction of convexity. More precisely we prove: any quasiconformal harmonic mapping between two …
It was shown by Kaup that every origin-preserving automorphism of quasi-circular domains is a polynomial mapping. In this paper, we study how the weight of quasi-circular domains and the degree of such automorphisms are related. By using the Bergman mapping, we prove that every origin-preserving automorphism of normal …
Study on regularity of optimal transport maps on convex domains with quadratic cost.
problem Regularity of optimal transport maps between convex domains with quadratic cost.
method Analysis of Cα-densities and C1,α boundary conditions, monotonicity formula for optimal transport maps. result Proves C1,1−ε-regularity for nondegenerate Cα-densities and C2,α-regularity for C1,α boundary. Study on images and singularities of pseudoholomorphic maps.
problem Characterize images and singularities of pseudoholomorphic maps.
method Analyzes pseudoholomorphic maps in domains and targets of dimension four.
result Proves properties of images and singularities of pseudoholomorphic maps.
The paper explores unique continuation properties for polyharmonic maps between Riemannian manifolds.
problem Investigating unique continuation principles for polyharmonic maps.
method Analyzing critical points of higher order functionals to prove extensions of known results in harmonic and biharmonic cases.
result Proving extensions of unique continuation principles for k-harmonic maps.
The Fock-Bargmann-Hartogs domain Dn,m(μ) (μ>0) in Cn+m is defined by the inequality ∥w∥2<e−μ∥z∥2, where (z,w)∈Cn×Cm, which is an unbounded non-hyperbolic domain in Cn+m. Recently, Yamamori gave an explicit formula for the Bergman kernel of the…
Study proper holomorphic maps between specific domains, proving rigidity under certain conditions.
problem Proper holomorphic maps between type-I irreducible bounded symmetric domains. method Analyzing maps under specific assumptions, using automorphisms and a defined map Gh. result Rigidity results for maps, showing conditions on dimensions and existence of automorphisms.
Optimal transport for functional data using Hilbert-Schmidt operators.
problem Optimal transport for distributions on function spaces with partially represented stochastic maps.
method Regularization technique to restrict transport maps to Hilbert-Schmidt operators, developing an efficient algorithm.
result Existence, uniqueness, and consistency of the Hilbert-Schmidt operator estimate for the transport map.
LRS-DAG improves domain adaptation for low-resource settings.
problem Maintaining performance on source domain after target domain adaptation.
method Adds encoder layers to map target to source domain, maintaining source performance.
result Outperforms fine-tuning on synthetic low-resource datasets.
Sourcerer uses deep learning to map land cover from limited labeled data.
problem Producing accurate land cover maps with scarce labeled data.
method Bayesian-inspired, deep learning approach with a novel regularizer.
result Sourcerer outperforms other methods, even with minimal labeled target data.
Hua domain, named after Chinese mathematician Loo-Keng Hua, is defined as a domain in Cn fibered over an irreducible bounded symmetric domain Ω⊂Cd(d<n) with the fiber over z∈Ω being a (n−d)-dimensional generalized complex ellipsoid Σ(z). In general, a Hua domain is a nonhom…
New method improves generative modeling on convex domains using regularized mirror maps and Student-t priors.
problem Challenges in generative modeling on convex domains with heavy-tailed targets.
method Mirror Flow Matching with regularized mirror maps and Student-t priors.
result Empirically outperforms baselines and achieves competitive sample quality.
Smoothly bounded domains have special functions that are plurisubharmonic.
problem Finding smooth functions that are plurisubharmonic on bounded domains.
method Proving existence of smooth defining functions that are p-plurisubharmonic. result Smooth domains with smooth p-convex boundaries admit smooth defining functions that are p-plurisubharmonic. For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from (Ω,g) to a compact Riemannian manifold (N,h)⊂Rk without boundary. We generalize the notion of stationary harmonic map and prove the partial regularity. We also discuss the global Li…
Selects points from Jordan domains on Riemannian surfaces.
problem Selecting points from Jordan domains in Riemannian surfaces.
method Fiber bundle theory and conformal mappings.
result Space of Jordan domains retracts onto round disks.
In this paper, we propose a novel learning framework for the problem of domain transfer learning. We map the data of two domains to one single common space, and learn a classifier in this common space. Then we adapt the common classifier to the two domains by adding two adaptive functions to it respectively. In the com…
The paper extends a method for numerical conformal mappings to surfaces using Laplace-Beltrami equations.
problem Computing conformal mappings between Riemannian surfaces.
method Adapting the conjugate function method to Riemannian surfaces using hp-adaptive finite element methods. result Highly accurate numerical computations of conformal mappings on surfaces, including complex geometries.
Novel OT framework for unsupervised domain translation.
problem Finding meaningful correspondence between two unpaired domains.
method Reformulate UDT in Optimal Transport framework, propose novel dynamical framework for efficient computation.
result Existence and uniqueness of optimal mappings proved for a large family of transport costs.
We study harmonic maps from Riemannian manifolds into arbitrary non-positively curved and CAT(-1) metric spaces. First we discuss the domain variation formula with special emphasis on the error terms. Expanding higher order terms of this and other formulas in terms of curvature, we prove an analogue of the Eels-Sampson…
The concept of a conformal deformation has two natural extensions: quasiconformal and harmonic mappings. Both classes do not preserve the conformal type of the domain, however they cannot change it in an arbitrary way. Doubly connected domains are where one first observes nontrivial conformal invariants. Herbert Groetz…
timeXplain bridges AI and time series, making predictions understandable.
problem Making time series classifier predictions interpretable.
method Developed a framework that combines time series data with model-agnostic explainers.
result timeXplain improves the interpretability of time series classifiers.
Framework translates unlabeled images between domains.
problem Translating unlabeled images between domains with no supervision.
method Skip-connected encoder-generator structure trained with GAN, cycle, and semantic consistency losses.
result Framework can learn semantic mappings for face images without supervised one-to-one mapping.
Complex domains covering manifolds are biholomorphic to balls.
problem Covering compact manifolds by bounded domains with smooth boundaries.
method Using biholomorphic mappings and properties of C1,1 boundaries. result Bounded domains with smooth boundaries covering compact manifolds are biholomorphic to the unit ball.
Study visibility properties of Kobayashi distance on unbounded domains.
problem Understanding visibility properties of Kobayashi distance on unbounded domains.
method Analyzing visibility properties in the context of Kobayashi hyperbolic domains, focusing on unbounded domains and their boundary behavior.
result Carathéodory-type extension theorem for biholomorphisms between planar domains, including infinitely-connected domains.
The paper resolves a problem about metric inequivalence and characterizes proper holomorphic maps.
problem Metric inequivalence and characterization of proper holomorphic maps.
method Explicit characterization of proper holomorphic maps from a finitely-connected planar domain onto the unit disk.
result Characterization of proper holomorphic maps from a finitely-connected planar domain onto the unit disk.
CMC method detects causal relationships in time series data.
problem Understanding causal relationships in nonlinear systems.
method Cross-Mapping Coherence method, based on nonlinear state-space reconstruction and coherence metrics.
result CMC accurately identifies causal connections in various systems.
In this paper, we are interested in the construction of quasiconformal mappings between domains of the Heisenberg group H that minimise a mean distortion functional. We propose to construct such mappings by considering a corresponding problem between domains of Poincaré half-plane H. The first map we construc…
Established a transport theorem for nonconvecting domains on an embedded manifold.
problem Transport theorems for nonconvecting domains evolving on an embedded manifold.
method Used geometric measure theory and the divergence theorem to prove the theorem.
result Proved a transport theorem for nonconvecting domains on an embedded manifold.
New findings on domains without parabolic minimal submanifolds and weakly hyperbolic domains.
problem Characterizing domains without parabolic minimal submanifolds and weakly hyperbolic domains.
method Analyzing properties of domains and their boundaries, using tubular neighborhoods and conformal harmonic maps.
result Domains without parabolic minimal submanifolds and weakly hyperbolic domains have specific geometric properties.
When learning a mapping from an input space to an output space, the assumption that the sample distribution of the training data is the same as that of the test data is often violated. Unsupervised domain shift methods adapt the learned function in order to correct for this shift. Previous work has focused on utilizing…
In 1996, Shi generalized the epsilon-regularity theorem of Schoen and Uhlenbeck to energy-minimizing harmonic maps from a domain equipped with a bounded measurable Riemannian metric. In the present work we prove a compactness result for such energy-minimizing maps. As an application, we combine our result with Shi's th…
Our goal is to provide a novel method of representing 2D shapes, where each shape will be assigned a unique fingerprint - a computable approximation to a conformal map of the given shape to a canonical shape in 2D or 3D space (see page 22 for a few examples). In this paper, we make the first significant step in this pr…
A totally geodesic map f:X1→X2 between Hermitian symmetric spaces is tight if its image contains geodesic triangles of maximal area. Tight maps were first introduced in [BIW09], and were classified in [Ham13, Ham14, HO14] in the case of irreducible domain. We complete the classification by analy…
Maximal spacetimes have unique past/future sets.
problem Characterizing maximal spacetimes.
method Developing map analysis and diamond properties.
result Maximal spacetimes have unique past/future sets.
CMA-ME combines CMA-ES and MAP-Elites for better quality and diversity in continuous domains.
problem Finding a diverse set of high-quality solutions in complex continuous domains.
method Combines CMA-ES self-adaptation with MAP-Elites archiving and mapping.
result CMA-ME outperforms MAP-Elites in both quality and diversity of solutions.