CuBAS selects informative data points based on curvature for better classification.
problem Lack of efficient sampling strategies for maximizing dataset informativeness.
method Information-geometric framework using curvature scores to select labeled data.
result Consistent and statistically significant improvements over random and uncertainty-based sampling.
We study the (2+2)-Einstein warped product manifolds, where the scalar curvature of the Base is a multiple of the warping function, and we called this condition (inside a warped product manifold) f-curvature-Base (RfB).The aim of this paper is to check if there are Base-manifolds with non-flat metrics that sa…
The paper introduces curvature-based clustering algorithms for graph analysis.
problem Identifying densely connected substructures in graphs for community detection.
method Discrete Ricci curvatures and geometric flows to reveal community structure.
result The curvature-based approach can identify overlapping communities in graphs.
Study proposes curvature flow model for Drosophila dorsal closure.
problem Modeling and understanding Drosophila dorsal closure during embryonic development.
method Curvature-based mathematical model, analysis of maximum-principle and integral-estimates, numerical approximation scheme.
result Established global existence and convergence for the model.
We prove that a Ricci curvature based method of triangulation of compact Riemannian manifolds, due to Grove and Petersen, extends to the context of weighted Riemannian manifolds and more general metric measure spaces. In both cases the role of the lower bound on Ricci curvature is replaced by the curvature-dimension co…
In this paper, we give the sharp upper bound for the number of vertices with positive curvature in a planar graph with nonnegative combinatorial curvature. Based on this, we show that the automorphism group of a planar---possibly infinite---graph with nonnegative combinatorial curvature and positive total curvature is …
We propose here a new discretization method for a class continuum gauge theories which action functionnals are polynomials of the curvature. Based on the notion of holonomy, this discretization procedure appears gauge-invariant for discretized analogs of Yang-Mills theories, and hence gauge-fixing is fully rigorous for…
Study on warped product Yamabe solitons with constant fiber curvature.
problem Characterizing nontrivial warped product Yamabe gradient solitons.
method Investigation of warped product manifolds, derivation of scalar curvature estimates.
result Nontrivial warped product Yamabe gradient solitons have constant scalar curvature in the fiber.
Persistent homology can recognize knotting in curves.
problem Recognizing knotting in curves
method Compute one-dimensional persistent homology, extract cycle representatives, and assign a hypergraph curvature-based score.
result Systematic differences between knotted and unknotted structures are revealed.
New method improves calibration of neural networks by targeting robust margins and local smoothness.
problem Poor calibration of neural networks, leading to unreliable confidence estimates.
method Intervene on training procedure by targeting robust margins and local smoothness.
result Improved out-of-sample calibration without sacrificing accuracy.
This paper connects graph curvature to community structure.
problem Understanding the relationship between network curvature and community formation.
method Defining curvature on networks and analyzing its relation to community structure.
result Apriori bounds on the curvature of intercommunity edges.
The paper classifies surfaces with constant skew curvature in 3-space forms.
problem Classifying surfaces with constant skew curvature in 3-space forms.
method Variational characterization and flow of binormal vector field.
result Classification of rotational surfaces with constant skew curvature.
We decrease the rms mean curvature and area of a variable surface with a fixed boundary by iterating a few times through a curvature-based variational algorithm. For a boundary with a known minimal surface, starting with a deliberately chosen non-minimal surface, we achieve up to 65 percent of the total possible decr…
Based on two classical notions of curvature for curves in general metric spaces, namely the Menger and Haantjes curvatures, we introduce new definitions of sectional, Ricci and scalar curvature for networks and their higher dimensional counterparts. These new types of curvature, that apply to weighted and unweighted, d…
Duality principle for approximation of geometrical objects (also known as Eudoxus exhaustion method) was extended and perfected by Archimedes in his famous tractate "Measurement of circle". The main idea of the approximation method by Archimedes is to construct a sequence of pairs of inscribed and circumscribed polygon…
New method calculates discrete curvature using effective resistances.
problem Calculating discrete curvature on graphs.
method Effective resistances to calculate curvature on graph nodes and links.
result Relation to established discrete curvatures and convergence to continuous curvature.
For the studied cases in [10], the author showed that having the {\textit {f-curvature-Base}} (RfB) is equal to requiring a flat metric on the base-manifold. In [11] the authors introduced a new kind of Einstein warped product manifold, composed by positive-dimensional manifold and negative-dimensional manifold…
New method connects curvature and Persistent Homology for networks.
problem Efficient computation of Persistent Homology for complex networks.
method Discrete Morse Theory, Bloch's extension, Forman-Ricci curvature.
result Efficient Persistent Homology scheme using curvature-based approach.
This paper reviews discrete curvature models for geometric data analysis.
problem Capturing intrinsic geometric structure in diverse data representations.
method Comprehensive review of discrete curvature models from Riemannian and metric geometry perspectives.
result Systematic pipeline for curvature-driven data analysis and learning.
Graphs can be smoothed or squashed too, study finds.
problem Graph Neural Networks struggle with over-smoothing and over-squashing issues.
method Unified framework using Ollivier-Ricci curvature to address both issues.
result Over-smoothing and over-squashing linked to positive and negative graph curvature respectively.
The paper studies 8D manifolds with a specific tensor field called a cubic discriminant.
problem Characterizing and understanding 8D Riemannian manifolds with reduced structure groups.
method Introducing an almost quaternion-Hermitian structure and a cubic discriminant tensor field.
result Only two non-flat, integrable examples of these structures are found: quaternion-Kähler symmetric spaces.
In a previous paper, we proved that a projective Kähler manifold of positive total scalar curvature is uniruled. At the other end of the spectrum, it is a well-known theorem of Campana and Kollár-Miyaoka-Mori that a projective Kähler manifold of positive Ricci curvature is rationally connected. In the present work, we …
Graph rewiring method alleviates over-squashing in GNNs.
problem Over-squashing and bottlenecks in graph neural networks.
method Curvature-based graph rewiring method.
result Negatively curved edges cause over-squashing in GNNs.
In the recent articles \cite{PSU1,PSU3}, a number of tensor tomography results were proved on two-dimensional manifolds. The purpose of this paper is to extend some of these methods to manifolds of any dimension. A central concept is the surjectivity of the adjoint of the geodesic ray transform, or equivalently the exi…
Paper explores entropic curvature in Markov chains, comparing it to other curvatures.
problem Comparing entropic curvature to other curvatures in Markov chains.
method Adapted Γ-calculus for θ-curvatures, explicit lower bounds, curvature perturbation.
result Entropic curvature differs significantly from other curvature notions.
HLRC offers a new curvature metric for hypergraphs that balances interpretability and efficiency.
problem Challenges in geometric characterization of hypergraphs with higher-order interactions.
method Hypergraph lower Ricci curvature (HLRC) defined in closed form.
result HLRC consistently reveals meaningful higher-order organization in diverse hypergraph datasets.
AdamQLR optimizes Adam with K-FAC heuristics, achieving comparable performance to tuned benchmarks.
problem Improving the performance of Adam optimizers with stabilizing heuristics.
method Combining Adam's update directions with K-FAC's heuristics (damping and learning rate selection).
result Untuned AdamQLR can achieve comparable performance to tuned benchmarks.
Paper proposes LCP for structural encodings, outperforming existing methods.
problem Improving Graph Neural Networks performance through effective structural encodings.
method Geometric perspective, Local Curvature Profiles (LCP) for structural encodings, combining with global positional encodings, comparing with rewiring techniques.
result LCP significantly outperforms existing structural encodings and combining LCP with global positional encodings improves performance.
The paper refines classical covariance asymptotics using geometric information geometry.
problem Deviation of finite-sample behavior from classical predictions in curved models.
method Develops a curvature-aware refinement by viewing parametric families as Riemannian manifolds with Fisher-Rao metric.
result Derives an \(n^{-2}\) correction to the leading \(n^{-1}I(θ)^{-1}\) covariance term for score-root estimators.
The paper explores almost Ricci solitons on Finsler spaces, proving conditions for their existence.
problem Characterizing almost Ricci solitons on Finsler measure spaces.
method Introducing and investigating gradient almost Ricci solitons, proving conditions for existence.
result Conditions for the existence of gradient almost Ricci solitons on Finsler measure spaces.
The paper proves conditions for positive scalar curvature metrics on manifolds with incompressible hypersurfaces.
problem Conditions for the existence of metrics with positive scalar curvature on manifolds with incompressible hypersurfaces.
method Analyzing surgeries and applying the positive mass theorem with incompressible conditions.
result Establishes positive mass theorem with incompressible conditions for specific manifolds.
This paper proposes low-complexity algorithms for finding approximate second-order stationary points (SOSPs) of problems with smooth non-convex objective and linear constraints. While finding (approximate) SOSPs is computationally intractable, we first show that generic instances of the problem can be solved efficientl…
Study on bending energy of surfaces with curvature concentration, deriving new lower bounds.
problem Analyzing the Willmore energy of surfaces with curvature concentration.
method Using isoperimetric inequalities and framed loops, derive new lower bounds for the bending energy.
result Optimal blowup rates of the Willmore energy when curvature is concentrated.
Sharp isoperimetric inequality derived from Q-curvature for smooth metrics.
problem Deriving a sharp isoperimetric inequality from the top order Q-curvature.
method Analyzing the relationship between Q-curvature and sectional curvature, applying isoperimetric inequality.
result Sharp isoperimetric inequalities derived for domains with smooth boundaries.
We consider the quantum completeness problem, i.e. the problem of confining quantum particles, on a non-complete Riemannian manifold M equipped with a smooth measure ω, possibly degenerate or singular near the metric boundary of M, and in presence of a real-valued potential V∈Lloc2(M). The main …
CurvSSL improves SSL by aligning local manifold curvature.
problem Improving self-supervised learning by capturing local manifold geometry.
method CurvSSL augments Barlow Twins with a curvature-based regularizer to align and decorrelate embeddings across augmentations.
result Curvature-regularized SSL yields competitive or improved linear evaluation performance.
We analyze the Hessian spectra of large models up to 100B parameters.
problem Accurate Hessian spectra of large foundation models are difficult to obtain.
method We use shard-local finite-difference Hessian vector products and stochastic Lanczos quadrature.
result We produce the first large-scale spectral density estimates of foundation models.
Automatic segmentation of auditory ossicles from CT images using Ricci curvature.
problem Automatic diagnosis of ossicles' diseases from 3D CT images of the head.
method Proposes a completely automatic method that locates and segments ossicles without manual labels or templates, using Ricci curvature in an energy function.
result Performance of the proposed method using discrete Forman-Ricci curvature is superior to state-of-the-art methods.
GeoTop resolves topological ambiguity in diagnostic imaging using geometric-topological analysis.
problem Topological equivalence between benign and malignant structures in diagnostic images.
method Combines Topological Data Analysis and Lipschitz-Killing Curvatures to resolve ambiguity.
result Achieves 3.6% accuracy improvement and reduces false positives/negatives by 15-18%.
Spectro-Riemannian Graph Neural Networks integrate spectral and curvature signals for better graph representation learning.
problem Enhance graph representation learning by leveraging spectral and curvature signals.
method Proposes Spectro-Riemannian Graph Neural Networks (CUSP) that combines spectral and curvature insights.
result Empirical evaluation shows CUSP outperforms state-of-the-art models by up to 5.3%.
New method uses curvature to improve graph neural networks.
problem Graph Neural Networks struggle with over-smoothing and over-squashing.
method Augmented Forman-Ricci curvature (AFRC) for scalable rewiring.
result AFRC effectively mitigates over-smoothing and over-squashing.
Novel symmetry found in nanocarbons' discrete principal curvature structure.
problem Identifying novel symmetries in nanocarbons' geometric structures.
method First-principles calculations and discrete geometry analysis.
result Discovery of a novel symmetry (pre-constant discrete principal curvature) in nanocarbons.
Enhanced Sampling Scheme improves masked generative modeling.
problem Limitations of existing sampling schemes in masked non-autoregressive generative modeling.
method ESS consists of three stages: Naive Iterative Decoding, Critical Reverse Sampling, and Critical Resampling.
result ESS achieves significant performance gains in unconditional and class-conditional sampling.
PRS improves rejection sampling by learning better proposals.
problem High rejection rate in traditional rejection sampling.
method PRS uses a kernel estimator to learn better sampling proposals.
result PRS guarantees a low number of accepted samples.
This paper reviews various sampling methods from statistics and machine learning.
problem Addressing sampling methods in statistics and machine learning.
method Explains and reviews simple random sampling, bootstrapping, stratified sampling, cluster sampling, multistage sampling, network sampling, snowball sampling, and sampling from cumulative distribution function.
result Summarizes characteristics, pros, and cons of different sampling methods.
RISA improves VFL by using imputed samples with low uncertainty.
problem Limited overlapping samples constrain VFL performance.
method Imputing non-overlapping samples and using evidence theory to select reliable imputed samples.
result Significant performance gains achieved, especially with limited overlapping samples.
Improved privacy-preserving methods for estimating multiple samples from distributions.
problem Estimating multiple samples from distributions while maintaining privacy.
method Developed new multi-sampling techniques for differentially private data estimation.
result Achieved significant reduction in sample complexity for multi-sampling from finite domains and Gaussian distributions.
Paper introduces a new sampling method combining Consistency Models with importance sampling.
problem Inherent errors in samples and high NFEs for high-quality samples in Boltzmann distributions.
method Combines Consistency Models with importance sampling to produce unbiased samples with minimal NFEs.
result Produces unbiased samples using only 6-25 NFEs, comparable to 100 NFEs for DDPMs.