CAD detects anomalies and selects prototypes using polyhedron curvature.
problem Anomaly detection and prototype selection in data.
method Curvature Anomaly Detection (CAD) and Kernel CAD approach using polyhedron curvature.
result The proposed methods are effective for anomaly detection and prototype selection.
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.
MCBP detects boundaries in high-dimensional data using curvature.
problem Boundary detection in high-dimensional data.
method MCBP uses mean curvature to model data manifold curvature.
result MCBP improves clustering performance in complex scenarios.
Unified piecewise-linear Ricci flows improve community detection.
problem Improving community detection in graph neural networks.
method Proposed piecewise-linear Ricci curvature flows with surgeries.
result Flow consistently outperforms baseline models on real-world datasets.
Persistent homology detects curvature from sampled points.
problem Understanding the geometric information encoded in short intervals of persistent homology.
method Persistent homology computations and average persistence landscapes.
result Persistent homology detects curvature of disks from sampled points.
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.
New method uses Ricci curvature for hypergraph clustering, outperforming existing techniques.
problem Community detection in hypergraphs with large hyperedges.
method Extending Ricci flow to hypergraphs by defining edge probability measures and transporting them on the line expansion.
result Enhanced sensitivity to hypergraph structure, especially in large hyperedges.
Geometric sampling of networks using curvature measures.
problem Sampling and analyzing complex network structures.
method Three types of discrete curvature (Forman-, full Forman-, Haantjes-Ricci) for edge-based and node-based sampling.
result Effective detection of networks' backbone and coarse structure.
Mapping complex input data into suitable lower dimensional manifolds is a common procedure in machine learning. This step is beneficial mainly for two reasons: (1) it reduces the data dimensionality and (2) it provides a new data representation possibly characterised by convenient geometric properties. Euclidean spaces…
Homological stability fails for 4-manifold moduli spaces, detected by new class.
problem Homological instability for moduli spaces of 4-manifolds.
method Used Seiberg-Witten equations to construct a new characteristic class.
result Homological discrepancy between smooth and topological 4-manifold moduli spaces.
Detect spacetime curvature with event causality measurements.
problem Detecting spacetime curvature without rulers and clocks.
method Prove spacetime non-flatness through causal relations.
result Sixteen measurements verify non-flatness of non-conformally flat spacetimes.
Detect spacetime curvature without rulers and clocks in 3D.
problem Detecting spacetime curvature without traditional measurement tools.
method Generalized results from 2D to 3D spacetime, proving well-stitched spacetime for conformally flat cases.
result A 3D spacetime is well-stitched if and only if it is conformally flat, providing a tool for curvature detection.
The space of graphs is often characterised by a non-trivial geometry, which complicates learning and inference in practical applications. A common approach is to use embedding techniques to represent graphs as points in a conventional Euclidean space, but non-Euclidean spaces have often been shown to be better suited f…
Detects underspecification in pre-trained models using local ensembles.
problem Underspecification in pre-trained models where many predictors are consistent with training data.
method Uses local second-order information to approximate prediction variance across an ensemble of models.
result Capable of detecting underspecification in pre-trained models on test data.
Study constructs solutions for evolving hypersurfaces using inverse spacetime mean curvature.
problem Evolution of hypersurfaces in spacetime.
method Weak solutions for hypersurfaces evolving along inverse spacetime mean curvature in asymptotically flat maximal initial data sets.
result Weak solution detects both future- and past-trapped apparent horizons.
If a given behavior of a multi-agent system restricts the phase variable to a invariant manifold, then we define a phase transition as change of physical characteristics such as speed, coordination, and structure. We define such a phase transition as splitting an underlying manifold into two sub-manifolds with distinct…
Single model detects abnormal samples across diverse tasks.
problem Detecting abnormal samples in machine learning.
method Introduced Diffusion Paths (DiffPath) using a single unconditional diffusion model.
result Single model performs OOD detection across diverse tasks.
We introduce a novel definition of curvature for hypergraphs, a natural generalization of graphs, by introducing a multi-marginal optimal transport problem for a naturally defined random walk on the hypergraph. This curvature, termed \emph{coarse scalar curvature}, generalizes a recent definition of Ricci curvature for…
Study detects non-trivial elements in diffeomorphism groups via trivalent graphs.
problem Detecting non-trivial elements in homotopy groups of diffeomorphism spaces.
method Using Kontsevich classes and trivalent graphs, we lift elements from one moduli space to another.
result Non-trivial elements in π∗(BDiff∂(Dd))⊗Q are lifted to π∗(BDiff⊔(DdimesI))⊗Q and π∗(M∂psc(Dd)h0)⊗Q. Formula calculates mass using cube faces and edges.
problem Measuring mass of 3-manifolds.
method Cube faces and edges, mean curvature, dihedral angle, geodesic curvature, angle defect.
result Mass formula connects to Gromov's theory and Gauss-Bonnet theorem.
Ricci flow smooths locally collapsing manifolds with controlled curvature.
problem Locally collapsing manifolds with controlled Ricci curvature.
method Ricci flow for a definite period of time, detecting collapsing infranil fiber bundles.
result Topological conditions detect collapsing infranil fiber bundles.
Study on nonspin manifolds with spin boundary, showing nonconnectedness and nontrivial fundamental group.
problem Understanding spaces of positive scalar curvature metrics on totally nonspin manifolds.
method Analysis of positive scalar curvature metrics on manifolds with spin boundary, using propagation techniques.
result Spaces of positive scalar curvature metrics are not connected and have nontrivial fundamental groups for certain dimensions.
In this paper we are dealing with mean curvature flow with surgeries of two-convex hypersurfaces. The main focus is to expand on the discussion in Section 3 of Mean Curvature Flow with Surgeries of Two-Convex Hypersurfaces by Huisken and Sinestrari. Firstly we wish to establish how the neck detection lemma allows us …
We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling the one of gravitational collapse. In particular, the mean curvature simulation of a one-parameter family of initial data reveals the exi…
Study of null mean curvature flow on de Sitter lightcone, related to 2d-Ricci flow.
problem Analyzing singularity formation and asymptotic behavior of null mean curvature flow.
method Rescaling procedure to relate to 2d-Ricci flow, singularity analysis, asymptotic behavior study.
result Ancient solutions to the flow can be understood in terms of 2d-Ricci flow.
New curvature equations obstruct integrability of complex structures.
problem Understanding curvature obstructions to integrability of complex structures.
method Direct approach using Nijenhuis tensor derivatives and curvature scalars.
result Certain complex structures cannot coexist with non-flat constant curvature metrics.
Curvature regularization prevents distortion in graph embeddings.
problem Graph topology patterns distort in Euclidean space, making detection difficult.
method Proposes curvature regularization to enforce flatness in embedding manifolds.
result Significant improvements in five embedding methods on open graph datasets.
Study of mean curvature flow on null hypersurfaces leading to MOTS.
problem Detecting marginally outer trapped surfaces (MOTS) in null hypersurfaces.
method Analysis of mean curvature flow on null hypersurfaces with mild conditions.
result Existence and convergence of mean curvature flow to MOTS.
New method recovers curvature from heat diffusion data.
problem Recovering Riemannian curvature from heat diffusion.
method Information-theoretic approach using relative entropy.
result Local curvature determined by heat diffusion.
Proposes manifold-based unsupervised anomaly detection for visual data.
problem Rare anomalies in unlabeled data.
method Constant curvature manifolds, hyperspherical Variational Auto-Encoders (VAE) with gyroplane layer.
result State-of-the-art results on visual anomaly benchmarks and histopathology.
Motivated by questions in detecting minimal surfaces in hyperbolic manifolds, we study the behavior of geometric flows in complete hyperbolic three-manifolds. In most cases the flows develop singularities in finite time. In this paper, we investigate the mean curvature flow in a class of complete hyperbolic 3-manifolds…
We show that the Einstein-Hilbert functional, as a functional on the space of Reeb vector fields, detects the vanishing Sasaki-Futaki invariant. In particular, this provides an obstruction to the existence of a constant scalar curvature Sasakian metric. As an application we prove that K-semistable polarized Sasaki mani…
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.
We study the filling invariants at infinity div_{k} for Hadamard manifolds defined by Brady and Farb in [Filling-invariants at infinity for manifolds of nonpositive curvature]. Among other results, we give a positive answer to the question they posed: whether these invariants can be used to detect the rank of a symmetr…
Recently Brendle-Huisken introduced a fully nonlinear flow G. Their aim was to extend the surgery algorithm of Huisken-Sinestrari, into the Riemannian setting. The aim of this paper is to go through the details on how to perform neck detection for a closed, embedded hypersurface M0 in Rn+1 undergoing…
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.
New method detects foliation enlargeability.
problem Detecting foliations with positive scalar curvature.
method Defining a foliation version of Rosenberg index element.
result Foliation version of Rosenberg index element is nonzero for enlargeable foliations.
Study of congestion in negative curvature manifolds using fair-division algorithms.
problem Estimating and predicting the size and location of congestion core in negative curvature manifolds.
method Introducing a novel fair-division algorithm to estimate congestion core.
result Demonstrated the effectiveness of fair-division algorithms in estimating congestion core.
Graph Ricci flow reveals hidden hierarchies in stock market correlations.
problem Detecting hidden structures in the complex stock market graph.
method Using graph Ricci curvature and flow techniques to analyze the NASDAQ 100 index.
result Algorithm detects hidden hierarchies, community behavior, and clustering in financial markets.
The aim of this paper is to associate a measure for certain sets of paths in the Euclidean plane R2 with fixed starting and ending points. Then, working on parameterized surfaces with a specific Riemannian metric, we define and calculate the integral of the length over the set of paths obtained as the image…
In this article, we show that a Finsler--Laplacian introduced previously can detect changes in the Finsler metric that the marked length spectrum cannot. We also construct examples of non-reversible Finsler metrics in negative curvature such that 4λ1>h2, where λ1 is the bottom of the L2-spectrum and h the…
Study introduces weak elastic energy for curves on Riemannian surfaces.
problem Detecting curvature of curves on Riemannian surfaces.
method Relaxation starting from inscribed geodesic polygonals, defined in normalized isothermal coordinates.
result Relaxed energy detects intrinsic second-order Sobolev regularity and agrees with geodesic curvature.
We construct series of examples of exotic smooth structures on compact locally symmetric spaces of noncompact type. In particular, we obtain higher rank examples, which do not support Riemannian metric of nonpositive curvature. The examples are obtained by taking the connected sum with an exotic sphere. To detect the c…
This paper addresses the issue of feature importance landscapes in complex images and proposes a regularisation technique to improve network performance.
problem Feature importance landscapes in complex images are not as uniform as assumed, affecting network performance.
method Developed the PILCRO objective to regularize weight configurations, making importance landscapes smoother and more data-driven.
result P-regularised networks have a flat importance landscape, train faster, and perform better in accuracy and robustness.
We construct non-trivial elements of order 2 in the homotopy groups π8j+1+∗Diff(D6,∂), for * congruent 1 or 2 modulo 8, which are detected by the "assembling homomorphism" (giving rise to the Gromoll filtration), followed by the alpha-invariant in KO∗=Z/2. These elements are constructed by means of Mor…
We describe explicitly the large volume isoperimetric regions of a natural class of asymptotically flat manifolds, in any dimension. These isoperimetric regions detect the mass and the center of mass of such manifolds when viewed as initial data sets for the Einstein equations in general relativity. Using the positivit…
Ricci-Filtration enhances retrieval-augmented generation rerankers for query-answer tasks by using discrete Ricci flow on graphs.
problem Improving retrieval-augmented generation rerankers for query-answer tasks.
method Discrete Ricci flow on graphs to evaluate structural importance of chunks.
result Ricci-Filtration outperforms baseline methods in accuracy, precision, recall, and F1 scores.
Conformally invariant functionals on the space of knots are introduced via extrinsic conformal geometry of the knot and integral geometry on the space of spheres. Our functionals are expressed in terms of a complex-valued 2-form which can be considered as the cross-ratio of a pair of infinitesimal segments of the knot.…