Study shows topological features improve time series classification.
problem Classifying stochastic processes with varying noise and sampling.
method Topological data analysis features compared to statistical and raw features.
result Topological features lead to better classification performance.
Topology aids in solving machine learning classification problems.
problem Machine learning classification problems.
method Classical topology applied to neural networks.
result Topology guides neural network architecture and training.
Introduces topological deep learning for neural network classification problems.
problem Classifying neural networks using minimal topological structures.
method Formalizes classification problems in a topological setting.
result Demonstrates conditions for the feasibility of classification problems in neural networks.
The main approaches for node classification in graphs are information propagation and the association of the class of the node with external information. State of the art methods merge these approaches through Graph Convolutional Networks. We here use the association of topological features of the nodes with their clas…
Generic smooth plane-to-plane map germs are topologically equivalent to cones of mappings of the circle. We carry out a complete topological classification of smooth stable mappings of the circle and show how this classification leads, via the result mentioned above, to a topological classification of finitely determin…
Machine learning reveals hidden features in knot classification.
problem Classifying the topology of closed curves.
method Investigating shortcut methods used by ML for knot classification.
result Developed a dataset and code to remove non-topological features.
The main purpose of this paper is to give a topological and symplectic classification of completely integrable Hamiltonian systems in terms of characteristic classes and other local and global invariants.
We give a complete topological classification of minimal surfaces in Euclidian three-space.
This paper classifies 13D manifolds based on Bazaikin spaces.
problem Classifying 13-dimensional manifolds based on Bazaikin spaces.
method Topological classification modelled on Aloff-Wallach spaces and Eschenburg spaces.
result Complete topological classification for 13D manifolds based on Bazaikin spaces.
A novel method extracts topological features from word embeddings for text classification.
problem High dimensional and noisy text representations in natural language processing.
method Persistent homology for topological data analysis on word embeddings.
result Topological features outperform conventional text mining features on long textual documents.
Paper uses topological data analysis for time series classification.
problem Classifying univariate time series data, especially physiological signals.
method Persistent homology for feature engineering, followed by machine learning.
result Higher accuracy achieved with fewer features compared to traditional methods.
In this paper we complete the classification of topological symmetry groups for complete graphs Kn by characterizing which Kn can have a cyclic group, a dihedral group, or a subgroup of Dm×Dm where m is odd, as its topological symmetry group.
We give a complete topological classification of germs of holomorphic foliations in the plane under rather generic conditions. The key point is the introduction of a new topological invariant called monodromy representation. This monodromy contains all the relevant dynamical information, in particular the projective ho…
Classifies equivariant vector bundles over toric manifolds.
problem Classifying vector bundles over toric manifolds.
method Klyachko-type classification over invariant affine charts.
result Generalizes Klyachko's classification of toric vector bundles.
Complete classification of rod complements in 3-torus using topology.
problem Classifying rod complements in the 3-torus.
method Topological arguments.
result Complete classification of all rod complements in the 3-torus.
This text is about geometric structures imposed by robust dynamical behaviour. We explain recent results towards the classification of partially hyperbolic systems in dimension 3 using the theory of foliations and its interaction with topology. We also present recent examples which introduce a challenge in the classifi…
The paper classifies topological holonomy groups in SO(3).
problem Classifying holonomy groups in SO(3). method Analyzing and categorizing groups based on their properties.
result Different types of holonomy groups in SO(3) are identified and classified. TOGL adds topological info to GNNs, improving graph and node classification.
problem Graph neural networks lack substructure awareness, especially cycles.
method Integrates global topological information using persistent homology.
result Improves predictive performance for graph and node classification.
Persistent homology enhances graph classification by capturing long-range graph properties.
problem Lack of formal assessment of persistent homology in graph learning.
method Brief introduction and theoretical discussion of persistent homology in graph context, followed by empirical analysis.
result Persistent homology improves graph classification, especially for data with prominent topological structures.
We investigate the classification of topological quandles on some simple manifolds. Precisely we classify all Alexander quandle structures, up to isomorphism, on the real line and the unit circle. For the closed unit interval [0,1], we conjecture that there exists only one topological quandle structure on it, i.e. t…
Paper explores applying TDA to text classification, improving model performance.
problem Applying TDA to text classification is challenging due to the complexity of text geometry.
method Used word embeddings and TF-IDF vectors to extract topological features from text.
result Topological features improve classification results, especially in ensemble models.
In [Tohoku Math. J. 62 (2010), 45--53] the second author showed that, except for a few cases, the order N of a cyclic group of self-homeomorphisms of a closed orientable topological surface Sg of genus g≥2 determines the group up to a topological conjugation, provided that N≥3g. The first author et al…
A novel method classifies wafer defects using topological data analysis.
problem Classifying defect patterns on semiconductor wafers for maintenance and yield management.
method Representing defect patterns as vectors using topological features from persistent homology.
result The method outperforms CNN in accuracy and efficiency, especially with limited data.
Novel TRI-GNN framework improves graph classification robustness.
problem Graph neural networks suffer from over-smoothing and vulnerability to graph perturbations.
method Integrates higher-order graph information via persistent homology and local graph structure learning.
result TRI-GNN outperforms state-of-the-art baselines on node classification tasks.
The paper deals with topologically trivial Legendrian knots in tight and overtwisted contact 3-manifolds. The first part contains a thorough exposition of the proof of the classification of topologically trivial Legendrian knots (i.e. Legendrian knots bounding embedded 2-disks) in tight contact 3-manifolds. This part w…
New classification of nonorientable 4-manifolds with specific fundamental groups.
problem Classifying nonorientable 4-manifolds with cyclic fundamental groups.
method Simple cut-and-paste construction, using results from Hambleton-Kreck-Teichner and Khan.
result Plausible classification of a large set of nonorientable 4-manifolds with cyclic fundamental groups of order 2p.
Researchers extend topological classification to knotted semimetals in 3D.
problem Classifying semimetals with knotted nodal lines.
method Using cohomology and Mayer-Vietoris sequence to account for nodal line semimetals with space-time inversion symmetry.
result Manifest proof of Weyl charge cancellation condition for Z_2 monopole charge.
We show that the topological classification and the smooth classification are generically the same for certain families of plane curves in a semi-local case(the double local case). Especially we give the normal form of transversely jointed two families of plane curves with second order contact at the envelope.
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%.
Study characterizes compact Einstein-type manifolds with boundary.
problem Characterize compact Einstein-type manifolds with nonempty boundary.
method Proved a sharp boundary estimate, obtained Hawking mass bounds, and provided a topological classification for the boundary.
result Obtained a gap result for compact Einstein-type manifolds with boundary.
New method confirms conjectures about specific Legendrian knots.
problem Classifying Legendrian knots of specific topological types.
method New method of comparing Legendrian knots with nontrivial symmetry group.
result Conjectures about specific Legendrian knots are confirmed.
Study confirms Chern's conjecture on compact Hessian manifolds and classifies their topologies.
problem Global topological constraints and structural properties of compact Hessian manifolds.
method Novel fibration and splitting theorems, Chern's conjecture, Hitchin systems, Cheng-Yau solution.
result Topological classification of complete Hessian surfaces and closed orientable Hessian 3-manifolds.
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.
Enhanced binary classifier uses Urysohn's Lemma of Topology.
problem Binary classification challenges.
method Utilizes Urysohn's Lemma of Topology to construct separating functions.
result Exceptional performance in numerical experiments (95% to 100%).
We prove a structure theorem for closed topological manifolds of cohomogeneity one; this result corrects an oversight in the literature. We complete the equivariant classification of closed, simply connected cohomogeneity one topological manifolds in dimensions 5, 6, and 7 and obtain topological characterizations…
Enhances MIL performance in scarce data scenarios using topological inductive biases.
problem Low performance of MIL in data-scarce scenarios.
method Incorporates topological inductive biases into MIL framework.
result Average performance improvements of 15.3% for synthetic datasets, 2.8% for benchmarks, and 5.5% for rare anemia classification.
In the present article we obtain classification results and topological obstructions for the existence of translating solitons of the mean curvature flow.
Paper classifies pseudomanifolds over stratified spaces.
problem Classifying pseudomanifolds over stratified spaces.
method Introducing locally standard T-pseudomanifolds and using characteristic data. result Locally standard T-pseudomanifolds over topological stratified pseudomanifolds are classified by their characteristic data. The topological classification of gradient like Morse-Smale vector fields and diffeomorphisms on 3-manifolds was obtained.
We prove that each non-separable completely metrizable convex subset of a Frechet space is homeomorphic to a Hilbert space. This resolves an old (more than 30 years) problem of infinite-dimensional topology. Combined with the topological classification of separable convex sets due to Klee, Dobrowoslki and Torunczyk, th…
Paper classifies structures on 5D manifolds with specific rank and conditions.
problem Classifying tangent distributions on 5D manifolds.
method Established necessary and sufficient topological condition for existence.
result Classification of structures up to homotopy as formal Cartan distributions.
We classify Legendrian knots of topological type 76 having maximal Thurston--Bennequin number confirming the corresponding conjectures of Chongchitmate--Ng.
Classifies uncolored bonded knots with up to 7 singularity points.
problem Classifying uncolored bonded knots with up to 7 singularity points.
method Generation of planar graphs, conversion into bonded knot diagrams, use of Yamada polynomial, and brute-force Reidemeister moves.
result Systematic classification of uncolored bonded knots with singularity number at most seven.
Topology classifies bipolar surfaces; they are not embedded.
problem Classifying bipolar surfaces in the 5-sphere.
method Topological classification and area bounds calculation.
result Bipolar surfaces are not embedded.
Free actions of finite groups on spheres give rise to topological spherical space forms. The existence and classification problems for space forms have a long history in the geometry and topology of manifolds. In this article, we present a survey of some of the main results and a guide to the literature.
This paper examines how graph topology affects adversarial attacks on vertex classification.
problem Adversarial attacks on vertex classification are vulnerable to graph topology changes.
method Examined two topological graph characteristics and their impact on adversary perturbation budgets.
result Training sets including high-degree vertices or those ensuring all unlabeled nodes have neighbors can significantly increase the adversary's perturbation budget.
This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in 2002. The book is divided into three parts: the first is devoted to hyperbolic geometry, the second to surfaces,…
New invariant for classifying 4-manifolds up to cobordism.
problem Classifying closed, oriented topological 4-manifolds up to s-cobordism. method Introducing a stable range invariant after stabilization by a fixed number of S2imesS2. result A new invariant for classifying 4-manifolds up to s-cobordism.