Study explores geometric structure and prior for beta-logistic distribution.
problem Understanding the geometric structure and prior distributions of the beta-logistic distribution.
method Exploring dual geometric structure and uncovering α-parallel prior. result The beta-logistic distribution admits an α-parallel prior for any real number α. Improved tensor rank learning for CPD models using a generalized hyperbolic prior.
problem Inaccurate tensor rank determination leads to overfitting or underfitting in CPD models.
method Introduced a generalized hyperbolic prior for automatic tensor rank learning in probabilistic CPD models.
result Significantly improved performance in learning both low and high tensor ranks, even for low SNR cases.
Proposes a method to improve hierarchical clustering using set-level structural priors.
problem Lack of supervision for non-leaf structure in hierarchical clustering.
method Introduces set-level structural priors for semi-supervised hyperbolic hierarchical clustering.
result Improves label consistency and similarity-based tree quality over baselines.
Study on lengths of random multicurves on hyperbolic surfaces.
problem Distribution of lengths of random multicurves on closed hyperbolic surfaces.
method Using Margulis' thesis and Mirzakhani's equidistribution theorem for horospheres.
result Distribution of lengths admits a polynomial density, with coefficients expressible in terms of intersection numbers of psi-classes.
APo-VAE generates text in hyperbolic space for better hierarchical representation.
problem Lack of hierarchical structure in Euclidean embeddings for natural language.
method Adversarial Poincare Variational Autoencoder (APo-VAE) in hyperbolic latent space.
result APo-VAE outperforms Euclidean VAEs in capturing latent language hierarchies.
Learning graph representations via low-dimensional embeddings that preserve relevant network properties is an important class of problems in machine learning. We here present a novel method to embed directed acyclic graphs. Following prior work, we first advocate for using hyperbolic spaces which provably model tree-li…
In this paper we propose a novel framework for the construction of sparsity-inducing priors. In particular, we define such priors as a mixture of exponential power distributions with a generalized inverse Gaussian density (EP-GIG). EP-GIG is a variant of generalized hyperbolic distributions, and the special cases inclu…
Let G=⟨A,B⟩ be a non-elementary two generator subgroup of the isometry group of H2, the hyperbolic plane. If G is discrete and free and geometrically finite, its quotient is a pair of pants and in prior work we produced a formula for the number of essential self intersections (ESIs) of a…
ManifoldMind uses adaptive-curvature probabilistic spheres for trustworthy recommendations in semantic hierarchies.
problem Sparse and abstract recommendation domains where users explore diverse conceptual paths.
method Adaptive-curvature probabilistic spheres, soft multi-hop inference, and curvature-aware semantic kernel.
result Superior NDCG, calibration, and diversity compared to baselines on public benchmarks.
Our main result is that for all sufficiently large x0>0, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field k and systole bounded below by x0 has density one within the set of all commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds wit…
This paper continues our exploration of homology cobordism of 3-manifolds using our recent results on Cheeger-Gromov rho-invariants associated to amenable representations. We introduce a new type of torsion in 3-manifold groups we call hidden torsion, and an algebraic approximation we call local hidden torsion. We cons…
The paper proves existence of horo-convex hypersurfaces in hyperbolic space with specific curvature conditions.
problem Existence of horo-convex hypersurfaces with prescribed shifted Gauss curvatures in hyperbolic space.
method Existence result obtained via standard degree theory based on a prior estimates for solutions to the prescribed shifted Gauss curvature equations.
result Existence of horo-convex hypersurfaces in hyperbolic space under certain conditions.
This work estimates edge weights of edge-reinforced random walks using observed data.
problem Statistical estimation of edge weights in edge-reinforced random walks.
method Proposes an estimator based on the generalized method of moments using the magic formula and hyperbolic Gaussian structure.
result Analyzes the sample complexity of the proposed estimator.
Introduces hyperbolic generalized framed surfaces and their properties.
problem None explicitly stated; focuses on introducing new geometric objects.
method Generalization of hyperbolic framed surfaces and curves.
result Established conditions for a surface to be a hyperbolic generalized framed base surface and explored their singularities.
Keeping a basic tenet of economic theory, rational expectations, we model the nonlinear positive feedback between agents in the stock market as an interplay between nonlinearity and multiplicative noise. The derived hyperbolic stochastic finite-time singularity formula transforms a Gaussian white noise into a rich time…
Improves stability in hyperbolic neural networks for complex data generation.
problem Numerical instability in hyperbolic neural networks hinders complex architecture development.
method Proposes a novel hyperbolic AE-GAN architecture with stable layers.
result Demonstrates state-of-the-art performance in generating complex data.
The problem of identifying geometric structure in heterogeneous, high-dimensional data is a cornerstone of representation learning. While there exists a large body of literature on the embeddability of canonical graphs, such as lattices or trees, the heterogeneity of the relational data typically encountered in practic…
The study explores weakly p-Kähler hyperbolic manifolds.
problem Generalization and application of weakly p-Kähler hyperbolic manifolds. method Investigation of generalizations and applications.
result Exploration of weakly p-Kähler hyperbolic manifolds. Survey on a nonlinear d'Alembertian from general relativity.
problem Understanding a new nonlinear operator from relativity.
method Review of a new distributional d'Alembertian, comparison estimates, and exact representation formulas.
result Control of the timelike cut locus through optimal transport.
Automorphism groups of infinitely-ended groups are acylindrically hyperbolic.
problem Characterizing the hyperbolicity of automorphism groups of infinitely-ended groups.
method Proving acylindrical hyperbolicity through group properties and automorphism analysis.
result Automorphism groups of infinitely-ended groups are acylindrically hyperbolic.
HyPV-LEAD detects cryptocurrency anomalies proactively, improving financial security.
problem Cryptocurrency anomalies like mixing, fraud, and pump-and-dump operations are hard to detect due to class imbalance and temporal volatility.
method HyPV-LEAD integrates lead time into anomaly detection through window-horizon modeling, Peak-Valley sampling, and hyperbolic embedding.
result HyPV-LEAD achieves a PR-AUC of 0.9624 on Bitcoin transaction data, significantly outperforming state-of-the-art methods.
Paper introduces a method to generate physically feasible dynamics with physical priors.
problem Challenges in generating physically feasible dynamics under physical priors.
method Seamlessly incorporates physical priors into diffusion-based generative models.
result Efficient generation of physically realistic dynamics across various physical phenomena.
Study on new hyperbolicity notions for non-Kähler manifolds and their deformations.
problem Analyzing new hyperbolicity notions for non-Kähler complex manifolds.
method Introducing and analyzing two new notions of hyperbolicity for compact complex non-Kähler manifolds, and studying their behavior under smooth modifications.
result Established openness results for p-HS hyperbolicity and p-Kähler hyperbolicity under holomorphic deformations. Sparsity-promoting priors have become increasingly popular over recent years due to an increased number of regression and classification applications involving a large number of predictors. In time series applications where observations are collected over time, it is often unrealistic to assume that the underlying spar…
Study shows Morse elements are common in acylindrically hyperbolic groups.
problem Understanding generic elements in acylindrically hyperbolic groups.
method Analyzing Morse elements and outer automorphisms.
result Morse elements are common in acylindrically hyperbolic groups.
Upper bounds for volumes of hyperbolic polyhedra and links are derived.
problem Finding upper limits for volumes of generalized hyperbolic polyhedra and links.
method Application of Belletti's theorem and analysis of polyhedra with triangular faces and trivalent vertices.
result Improved upper bounds for volumes of hyperbolic polyhedra and links are derived.
GOAT improves attention mechanisms by learning better priors.
problem Standard attention mechanisms use a naive uniform prior, limiting flexibility and generalization.
method GOAT introduces a trainable, continuous prior that replaces the uniform assumption, maintaining compatibility with optimized kernels.
result GOAT avoids representational trade-offs and learns an extrapolatable prior that combines positional flexibility with length generalization.
We study conservative partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds. We show that they are always accessible and deduce as a result that every conservative C1+ partially hyperbolic in a hyperbolic 3-manifold must be ergodic, giving an afirmative answer to a conjecture of Hertz-Hertz-Ures in the co…
Develops generic spike-and-slab priors for high-dimensional linear regression.
problem Bayesian high-dimensional linear regression challenges.
method Proposes a class of generic spike-and-slab priors and a unified framework for theoretical assessment.
result Achieves nearly-optimal posterior contraction rate and model selection consistency under general conditions.
We study the contact geometry of scalar second order hyperbolic equations in the plane of generic type. Following a derivation of parametrized contact-invariants to distinguish Monge-Ampere (class 6-6), Goursat (class 6-7) and generic (class 7-7) hyperbolic equations, we use Cartan's equivalence method to study the gen…
Generalizes global hyperbolicity to higher signatures and proves compactness.
problem Global hyperbolicity in higher pseudo-Riemannian signatures.
method Generalization and proof of compactness of causal diamonds.
result Existence of solutions to a Plateau problem and characterization of holonomies.
Sharp asymptotics derived for phase retrieval and compressed sensing with random generative priors.
problem Phase retrieval and compressed sensing with random measurement matrices.
method Sharp asymptotics derived for optimal performance and polynomial algorithm for random generative priors.
result Compressed phase retrieval becomes tractable with random generative priors, unlike sparse priors.
Proposes new priors for neural networks to improve generalization and uncertainty.
problem Improving generalization and uncertainty estimation in neural networks.
method Exploits scalable and structured posteriors as priors with generalization guarantees.
result Improves generalization and uncertainty estimation with non-vacuous bounds.
Improved VAEs by training a contrastive prior to match posterior.
problem Prior hole problem in VAEs, leading to poor image generation.
method Introduced a contrastive energy-based prior and trained it using noise contrastive estimation.
result Significant improvement in VAE generative performance on various datasets.
A generalized hyperbolic tetrahedra is a polyhedron (possibly non-compact) with finite volume in hyperbolic space, obtained from a tetrahedron by the polar truncation at the vertices lying outside the space. In this paper it is proved that a volume formula for ordinary hyperbolic tetrahedra devised by J. Murakami and M…
We consider the wave equation on a product cone and find a joint asymptotic expansion for solutions near null and future infinities. The rates of decay seen in the expansion at future infinity are the resonances of a hyperbolic cone and were computed by the authors in a previous paper. The expansion treats an asymptoti…
Introduces foundation priors for using model-generated data in empirical research.
problem Using model-generated data as real observations in empirical research.
method Introduces foundation priors as an exponential-tilted, generalized Bayesian update of the user's primitive prior.
result Synthetic data reflects both model patterns and user's priors, enabling principled use in empirical work.
Asymmetric expansion preserves convexity in hyperbolic geometry.
problem Maintaining convexity in hyperbolic geometry under asymmetric expansions.
method Generalizing earlier results on radial expansion to asymmetric expansion.
result Asymmetric expansion of hyperbolic convex sets remains convex.
Improves AI-prior reliability for Bayesian inference.
problem Error propagation from predictive models into posterior inference.
method Rectified AI-informed prior elicitation framework.
result Significant reduction in bias and improvement in predictive performance.
Hyperbolic GANs improve image generation metrics.
problem Improving image generation quality in neural networks.
method Integrating hyperbolic layers into GAN architectures.
result Hyperbolic GANs achieve better metrics than Euclidean counterparts.
Study geodesic trees and exceptional directions in FPP on hyperbolic groups.
problem Understanding the geometry and uniqueness of geodesics in FPP on hyperbolic groups.
method Analyzing random geodesic trees and exceptional directions in the context of FPP on hyperbolic groups.
result The set of exceptional directions has strictly smaller Hausdorff dimension than the boundary, and hence has measure zero.
Generative models improve inverse problems by providing tailored priors.
problem Analyzing the error in inverse problems solved with generative priors.
method Quantitative error bounds for minimum Wasserstein-2 generative models.
result The error in the posterior due to the generative prior is bounded by the prior's error in Wasserstein-1 distance.
Complex hyperbolic manifolds with many totally geodesic submanifolds are arithmetic.
problem Characterizing arithmeticity of complex hyperbolic manifolds with certain submanifolds.
method Developing superrigidity theorems for complex hyperbolic lattices and proving nonexistence of certain maps.
result Finite volume complex hyperbolic n-manifolds containing infinitely many maximal totally geodesic submanifolds of dimension at least two are arithmetic. The study of orthospectrum and simple orthospectrum of hyperbolic surfaces.
problem Characterizing hyperbolic surfaces through their orthospectrum and simple orthospectrum.
method Analyzing the orthospectrum and simple orthospectrum of compact hyperbolic surfaces with geodesic boundary.
result Generic surfaces are determined by their orthospectrum and simple orthospectrum, and there are finitely many surfaces sharing the same orthospectrum and simple orthospectrum.
Study of weakly Kähler hyperbolic manifolds, proving Lang and Green-Griffiths conjectures.
problem Verifying Lang and Green-Griffiths conjectures for a new class of manifolds.
method Introducing weakly Kähler hyperbolic manifolds and investigating their spectral properties.
result Proving weakly Kähler hyperbolic manifolds are of general type and verifying various aspects of Lang and Green-Griffiths conjectures.
For any finitely generated, non-elementary, torsion-free group G that is hyperbolic relative to P, we show that there exists a group G∗ containing G such that G∗ is hyperbolic relative to P and G is not relatively quasiconvex in G∗. This generalizes a result of I. Kapovich for hyperbo…
Generative models produce realistic objects in many domains, including text, image, video, and audio synthesis. Most popular models---Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs)---usually employ a standard Gaussian distribution as a prior. Previous works show that the richer family of pri…
We introduce a generalization of the Dijkgraaf-Witten invariants for cusped or compact oriented 3-manifolds. We show that the generalized DW invariants distinguish some pairs of cusped hyperbolic 3-manifolds with the same hyperbolic volumes and with the same Turaev-Viro invariants. We also present an example of a pair …