Study distance functions on manifolds linking geometry to topology.
problem Understanding the relationship between curvature and topology on manifolds.
method Analyzing distance functions and their connection to manifold geometry.
result Alternative proofs of theorems linking curvature and topology.
Study uses equivariant topology to measure distances between G metric spaces.
problem Measuring distances between G metric spaces.
method Equivariant topology methods to derive lower bounds.
result Sharp bounds on Gromov Hausdorff distance between spheres.
Distance function to a finite set is a topological Morse function.
problem Characterizing the topological Morse function of a finite set.
method Analyzing the distance function to a finite set in \(\mathbb{R}^n\).
result Distance function is a topological Morse function, with precise critical points and indices.
Proposes Topology Distance for evaluating GANs.
problem Challenges in evaluating GANs' goodness.
method Builds Vietoris-Rips complex on image features and defines TD based on latent manifold comparisons.
result Demonstrates TD's superiority over existing metrics.
Defines contact surgery distance and shows it's bounded by topological surgery distance by 5.
problem Comparing contact structures on 3-manifolds.
method Defining contact surgery distance and proving upper bound on its value.
result Contact surgery distance is at most 5 larger than topological surgery distance.
New topologies for star-shaped sets without boundedness.
problem Defining convergence for unbounded star-shaped sets.
method Introducing radial distance functionals and new topologies.
result New topologies τWr and τAWr for star-shaped sets. Topology helps estimate chromatic numbers of random graphs on spheres.
problem Estimating chromatic numbers of random graphs on spheres.
method Topology, specifically connectivity of Lóvasz's neighborhood complex.
result Connectivity bound is useful in dimensions 1 and 2, but generally poor.
A new method clusters complex networks using topological and geometric structure.
problem Clustering complex networks with intricate topology.
method Centroid-based clustering strategy using Wasserstein distance and barycenter for persistence barcodes.
result Demonstrated effectiveness on simulated and real-world networks.
Study shows effective resistance distance yields more accurate network barycenter than Hamming distance.
problem Identifying the best metric for computing the Fréchet mean network.
method Compared the effectiveness of Hamming distance and effective resistance distance in capturing network topology.
result Effective resistance distance produces a more accurate Fréchet mean network.
The paper connects geometric and topological concepts to bound distances between metric spaces.
problem Bounding distances between metric spaces using Gromov-Hausdorff distance.
method Using Borsuk-Ulam theorems and Vietoris-Rips complexes, the paper obstructs the existence of certain continuous maps between complexes to bound discontinuities of functions.
result The paper provides new bounds on Gromov-Hausdorff distances between spheres of different dimensions.
Unlike the case of surfaces of topologically finite type, there are several different Teichmüller spaces that are associated to a surface of topological infinite type. These Teichmüller spaces first depend (set-theoretically) on whether we work in the hyperbolic category or in the conformal category. They also depend, …
For localization and mapping of indoor environments through WiFi signals, locations are often represented as likelihoods of the received signal strength indicator. In this work we compare various measures of distance between such likelihoods in combination with different methods for estimation and representation. In pa…
New method robustifies topological data analysis against outliers.
problem Outliers make topological data analysis unstable.
method Proposed a robust distance function (MoM Dist) for persistent homology.
result MoM Dist sublevel filtrations and weighted filtrations are consistent estimators in adversarial settings.
A new model encodes distances and topology in latent variables.
problem Modeling dissimilarity data with latent variables and invariances.
method Isometric Gaussian Process Latent Variable Model using Riemannian geometry and variational inference.
result The model can encode invariances in learned manifolds.
Study on Frechet distance properties for paths and graphs.
problem Understanding topological properties of Frechet distance spaces.
method Proving path-connectedness of Frechet distance spaces and metric balls.
result Spaces of paths and graphs under Frechet distance are path-connected.
It is commonly known that in Riemannian and sub-Riemannian Geometry, the metric tensor on a manifold defines a distance function. In Lorentzian Geometry, instead of a distance function it provides causal relations and the Lorentzian time-separation function. Both lead to the definition of the Alexandrov topology, which…
This work incorporates topological features via persistence diagrams to classify point cloud data arising from materials science. Persistence diagrams are multisets summarizing the connectedness and holes of given data. A new distance on the space of persistence diagrams generates relevant input features for a classifi…
Distance, normals, and double normals for real plane curves with singularities
problem Relation between normals and double normals and critical points of the squared distance function for real algebraic curves with singularities
method Investigate the topological discriminant of the distance function
result The topological discriminant consists of the evolute and distinguished normal lines at algebraic singular points
A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. In this paper, we define naturally extended mappings of distance-squared functions, wherein each component is a distance-squared function. We investigate the properties of these mappin…
Paper compares dimension reduction methods using topological analysis on EEG data.
problem Comparing dimension reduction methods on EEG data.
method Topological data analysis, including persistent homology, Wasserstein distance, and hypothesis tests.
result Different dimension reduction methods show significant qualitative differences across topological homologies.
A hierarchical clustering algorithm for data clouds without structure assumptions.
problem Exploring data clouds without making structure assumptions.
method Hierarchical topological clustering algorithm that infers persistence of outliers and clusters of arbitrary shape from data hierarchy.
result The algorithm can provide meaningful clusters in complex datasets.
The paper characterizes global hyperbolicity in Lorentzian manifolds without relying on manifold topology.
problem Characterizing global hyperbolicity in smooth Lorentzian manifolds without assuming manifold topology.
method Two formulations of global hyperbolicity: one using chronological diamonds and the other using properties of the Lorentzian distance function.
result The second formulation is equivalent to the definition of `Lorentzian metric space' and introduces the concept of d-reflectivity. Framework for analyzing dynamic topological changes in point clouds using persistent homology and dynamic optimal transport.
problem Analyzing transient structural reorganizations during dynamic phase transitions in time-evolutionary point clouds.
method Hierarchical dynamic evaluation framework driven by topological and hypergraph reconstruction strategy.
result Combining transport-based alignment with multi-scale entropy diagnostics for dynamic topological analysis.
A new method detects small holes in noisy data.
problem Detecting small holes in high-density regions from noise.
method Robust Density-Aware Distance (RDAD) filtration, incorporating distance-to-measure concept.
result The RDAD filtration prolongs the persistences of small holes, making them distinguishable from noise.
This work introduces a method to compare sparse neural network topologies using graph theory.
problem Comparing and understanding sparse neural network topologies, especially during training.
method Introducing Neural Network Sparse Topology Distance (NNSTD) to measure distances between different sparse neural networks.
result Sparse neural networks can outperform over-parameterized models without further structure optimization.
PhyloVAE learns tree topologies without supervision.
problem Learning accurate tree representations from evolutionary data.
method Unsupervised learning via variational autoencoders with efficient tree generation.
result PhyloVAE generates high-resolution tree topologies efficiently.
The L2-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type M in a Riemannian manifold (N,g) induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for…
For any pseudo-Anosov diffeomorphism on a closed orientable surface S of genus greater than one, it is known by the work of Bers and Thurston that the topological entropy agrees with the translation distance on the Teichmüller space with respect to the Teichmüller metric. In this paper, we consider random walks on th…
Paper introduces DP TDA for near-optimal private persistence diagrams.
problem Challenges in privatizing topological data analysis.
method Sensitivity analysis of persistence diagrams, use of exponential mechanism.
result Proposes near-optimal privacy mechanism for TDA.
Study shows horofunction compactification's topology matches dual norm's unit ball.
problem Global topology of horofunction compactification of Finsler manifolds.
method Construct explicit homeomorphisms for various spaces.
result Horofunction compactification homeomorphic to dual norm's unit ball.
A new algorithm reduces the size of datasets for TDA.
problem Processing large datasets with high dimensions in TDA is computationally infeasible.
method Introduced Characteristic Lattice Algorithm (CLA) for data reduction.
result CLA reduces dataset size while preserving geometric and topological features.
We study the topological types of pants decompositions of a surface by associating to any pants decomposition P, in a natural way its pants decomposition graph, Γ(P). This perspective provides a convenient way to analyze the maximum distance in the pants complex of any pants decomposition to a pants decomposition c…
Bounds and constructions for Gromov-Hausdorff distance between spheres.
problem Calculating distances between spheres using Gromov-Hausdorff metric.
method Explicit constructions and topological ideas based on Borsuk-Ulam theorem.
result Lower bounds are tight for specific cases of sphere distances.
We investigate a type of distance between triangulations on finite type surfaces where one moves between triangulations by performing simultaneous flips. We consider triangulations up to homeomorphism and our main results are upper bounds on distance between triangulations that only depend on the topology of the surfac…
This paper explores infinite-dimensional Teichmüller spaces and their properties.
problem Teichmüller spaces of infinite-type surfaces are complex and depend on base structures.
method Study various distance functions and Teichmüller spaces associated with infinite-type surfaces.
result Finitely supported Teichmüller space is dense in asymptotically isometric Teichmüller space.
This work introduces novel methods to identify and compare cycles across topological objects.
problem Identifying and comparing topological features, particularly cycles, across different topological objects.
method Two complementary approaches: dendrogram-based merge-tree algorithms and Stratified Gradient Sampling.
result Transformed cycle matching into hierarchical clustering and topological optimization framework.
Persistence diagrams (PDs) play a key role in topological data analysis (TDA), in which they are routinely used to describe topological properties of complicated shapes. PDs enjoy strong stability properties and have proven their utility in various learning contexts. They do not, however, live in a space naturally endo…
A new topology design improves zero-shot classification performance in contrastive learning.
problem Improving zero-shot classification performance in contrastive visual-textual alignment.
method Proposed an alternative topology design using multiple class tokens and an oblique manifold with negative inner product.
result Improves zero-shot classification performance by an average of 6.1%.
Study extends null distance concept to Lorentzian length spaces for spacetime analysis.
problem Understanding spacetime convergence and topology in Lorentzian geometry.
method Extend null distance concept to Lorentzian length spaces, study Gromov-Hausdorff convergence.
result First results on compatibility of null distance with synthetic curvature bounds in warped product Lorentzian length spaces.
We produce examples of codimension one foliations of the Euclidean and hyperbolic planes with bounded geometry which are topologically products, but for which leaves are non-recursively distorted. That is, the function which compares intrinsic distances in leaves with extrinsic distances in the ambient space grows fast…
This paper optimizes portfolios using TDA and financial news sentiment.
problem Effective portfolio diversification through understanding asset similarity.
method Integrates TDA with FinBERT sentiment scores for dynamic rebalancing.
result Outperforms traditional methods in returns and risk-adjusted performance.
Auto-encoders are among the most popular neural network architecture for dimension reduction. They are composed of two parts: the encoder which maps the model distribution to a latent manifold and the decoder which maps the latent manifold to a reconstructed distribution. However, auto-encoders are known to provoke cha…
Topological theory for qLDPC codes enables non-Clifford gates and magic state injection.
problem Fault-tolerant quantum computation in qLDPC codes with non-Clifford gates and magic state resources.
method Developed a topological theory using simplicial or CW complex structures and deformation retraction.
result Achieved non-Clifford gates and magic state injection in qLDPC codes with constant rate and polynomial distance.
In a recent work I showed that the family of smooth steep time functions can be used to recover the order, the topology and the (Lorentz-Finsler) distance of spacetime. In this work I present the main ideas entering the proof of the (smooth) distance formula, particularly the product trick which converts metric stateme…
The study finds pairs of curves at distance 5 in surface curve graphs.
problem Finding pairs of curves at distance 5 in the curve graph of closed surfaces.
method Applying Dehn twists to fixed curves and characterizing conditions for distance 5.
result Characterization of pairs of curves at distance 5 in surface curve graphs.
Defines pants distance for knotted surfaces in 4-manifolds.
problem Complexity of knotted surfaces in 4-manifolds.
method Defining pants distance and proving standard form conditions.
result Determines conditions for standard form of knotted surfaces.
The study bounds distances in simplicial complexes and defines new invariants for 3-manifolds and handlebody-knots.
problem Estimating distances in simplicial complexes associated with low-dimensional manifolds.
method Obtained bounds on distances in simplicial complexes using topological conditions on vertices and curve complexes. Defined new invariants for 3-manifolds and handlebody-knots using splitting distances.
result Splitting distances in simplicial complexes are bounded from below under stabilizations, leading to converging invariants.
This paper introduces a new metric for deep learning networks based on their classification performance.
problem The mystery and black-box nature of deep learning networks.
method Proposes a new distance measure based on the probabilistic performance of deep learning networks.
result The proposed metric space is compact and coincides with the quotient topological space.