Proposes a method to learn representations of higher-dimensional simplicial complexes.
problem Lack of methods for representing entire simplicial complexes.
method Geometric message passing schemes for end-to-end learning of simplicial complex representations.
result First method for learning representations of entire simplicial complexes.
Simplicial learning improves classification by generating compact sparse representations.
problem Difficulty in distinguishing classes on the same subspace.
method Evolutionary simplicial learning approach to sparse representations.
result Evolutionary simplicial learning outperforms other methods in multi-class classification.
Mixes higher-order simplicial complexes for data augmentation.
problem Lack of labeled data for complex systems with multiway interactions.
method Proposes mixup mechanisms for simplicial complexes, including linear and nonlinear mixup, and a convex clustering mixup.
result Synthetic simplicial complexes interpolate between existing data based on homomorphism densities.
Topo-MLP learns network representations without message passing.
problem Lack of efficient higher-order network modeling methods.
method Proposes Topo-MLP, a simplicial neural network algorithm using MLP and HONC loss.
result Demonstrates improved robustness and efficiency in representation learning.
Unified probabilistic foundation for fuzzy simplicial sets in dimensionality reduction.
problem Lack of clear probabilistic interpretation in fuzzy simplicial sets.
method Introducing a probabilistic framework explaining fuzzy simplicial sets as marginals of probability measures on simplicial sets.
result Unified probabilistic theoretical foundation for fuzzy simplicial sets.
Study of p-adic simplicial volumes and their properties.
problem Understanding simplicial volumes over p-adic seminormed rings. method Definition and study of p-adic simplicial volumes, homology bounds, and computation of volumes for surfaces. result Established homology bounds and computed p-adic simplicial volumes for surfaces. Integral foliated simplicial volume is a version of simplicial volume combining the rigidity of integral coefficients with the flexibility of measure spaces. In this article, using the language of measure equivalence of groups we prove a proportionality principle for integral foliated simplicial volume for aspherical m…
We define the ideal simplicial volume for compact manifolds with boundary. Roughly speaking, the ideal simplicial volume of a manifold M measures the minimal size of possibly ideal triangulations of M "with real coefficients", thus providing a variation of the ordinary simplicial volume defined by Gromov in 1982, t…
We consider the relation between simplicial volume and two of its variants: the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space. We show that integral foliated s…
BScNets expands graph learning to higher-order interactions.
problem Link prediction on graphs with higher-order interactions.
method Proposes a new SNN using block Hodge-Laplacian and block simplicial complexes.
result BScNets outperforms state-of-the-art models in link prediction.
We show that non-elliptic prime 3-manifolds satisfy integral approximation for the simplicial volume, i.e., that their simplicial volume equals the stable integral simplicial volume. The proof makes use of integral foliated simplicial volume and tools from ergodic theory.
We study the multiscale simplicial flat norm (MSFN) problem, which computes flat norm at various scales of sets defined as oriented subcomplexes of finite simplicial complexes in arbitrary dimensions. We show that the multiscale simplicial flat norm is NP-complete when homology is defined over integers. We cast the mul…
We analyze oversquashing in topological message-passing using relational structures.
problem Oversquashing in topological message-passing remains understudied.
method A unifying axiomatic framework that bridges graph and topological message-passing.
result Potential to advance topological deep learning.
Alexander's conjecture extended to infinite simplicial complexes.
problem Alexander's conjecture for infinite simplicial complexes.
method Generalization of recent result for finite simplicial complexes.
result Alexander's conjecture holds for infinite simplicial complexes.
Local Kan conditions enable differentiation of simplicial manifolds.
problem Differentiating simplicial manifolds into Lie algebroids.
method Expanding a technique for higher Lie groupoids to simplicial manifolds.
result Derivation of a method to differentiate simplicial manifolds into higher Lie algebroids.
Study simplicial volume in fiber bundles with connected groups.
problem Understanding simplicial volume in fiber bundles.
method Investigated fiber bundles with connected structure groups.
result Simplicial volume of total space matches trivial bundle under certain conditions.
The simplicial volume introduced by Gromov provides a topologically accessible lower bound for the minimal volume. Lafont and Schmidt proved that the simplicial volume of closed, locally symmetric spaces of non-compact type is positive. In this paper, we present a generalization of this result to certain non-compact lo…
Paper defines and evaluates DR complex for persistent homology.
problem Computing persistent homology of Euclidean point cloud data.
method Delaunay-Rips complex construction for speed and stability.
result DR produces stable persistence diagrams under point cloud perturbations.
The study limits how many parts regular simplicial partitions can overlap.
problem Bounding the intersection number of regular simplicial partitions.
method Analyzing the properties of regular simplicial partitions.
result Established a maximum limit for the intersection number.
The study examines conditions for minimal volume entropy of simplicial complexes.
problem Conditions for minimal volume entropy of simplicial complexes.
method Topological conditions and growth of fundamental groups.
result Examples of simplicial complexes with zero simplicial volume and large minimal volume entropy.
We provide sharp lower bounds for the simplicial volume of compact 3-manifolds in terms of the simplicial volume of their boundaries. As an application, we compute the simplicial volume of several classes of 3-manifolds, including handlebodies and products of surfaces with the interval. Our results provide the firs…
TopoNTK kernel captures higher-order interactions in simplicial complexes.
problem Graph neural networks miss higher-order interactions in relational systems.
method Introduces TopoNTK, an infinite-width kernel for simplicial message passing.
result TopoNTK captures topology invisible to graph kernels, improving expressivity and interpretability.
Positive simplicial volume implies locally symmetric space structure.
problem Understanding simplicial volume in locally homogeneous spaces.
method Analyzing properties of locally homogeneous Riemannian manifolds.
result Closed locally homogeneous manifolds with positive simplicial volume are locally symmetric.
Formula connects foliated simplicial volume with group cost.
problem Calculating integral foliated simplicial volume.
method Ergodic decomposition formula for simplicial volume.
result Integration formula linking foliated simplicial volume and group cost.
Extends circle pattern theorem to quasi-simplicial triangulations.
problem Characterize circle patterns on quasi-simplicial triangulated surfaces.
method Use finite covering technique to reduce problem to simplicial case, prove characterization by KAT inequalities.
result Curvature image is characterized by KAT inequalities.
Paper studies spaces of flattenings of simplicial spheres and their homotopy type.
problem Existence and uniqueness of differentiable structures on simplicial spheres.
method Analyzes spaces of flattenings of simplicial spheres and their homotopy type.
result Spaces of flattenings have the homotopy type of the orthogonal group.
We prove that cubical simplicial volume of oriented closed 3-manifolds is equal to one fifth of ordinary simplicial volume.
Study shows simplicial volume of certain fiber bundles is zero.
problem Understanding simplicial volume in fiber bundles with nonpositive curvature.
method Proved simplicial volume is zero for specific fiber bundles.
result Simplicial volume of certain fiber bundles is zero.
Constructs simplified or complexified simplicial complexes.
problem Efficiently simplifying or complexifying complex spaces.
method Embeddings of simplicial complexes into a simplicial ball with bounded degrees and low volume.
result Realizes complicated spaces as parts of a ball/sphere or gives spheres specific metrics.
Minimal volume entropy vanishes or is positive under certain fiber growth conditions.
problem Conditions for minimal volume entropy to be zero or positive.
method Analyzes topological conditions related to fiber growth of maps.
result Examples of finite simplicial complexes with zero simplicial volume and large minimal volume entropy.
Study simplicial volume of manifolds from reflection group trick.
problem Characterize manifolds with positive simplicial volume.
method Define a partial order on triangulations and solve explicitly for minimal elements.
result Explicitly solved triangulations of the two-dimensional sphere and performed extensive analysis for three-dimensional case.
Integral foliated simplicial volume is zero for certain amenable covers.
problem Calculating the integral foliated simplicial volume of specific manifolds.
method Using open amenable covers and fixed price property of fundamental groups.
result Integral foliated simplicial volume is zero for manifolds with multiplicity at most n.
The paper proves simplicial volume positivity for certain nonpositively curved 4-manifolds with nonzero Euler characteristic.
problem Proving positivity of simplicial volume for specific types of 4-manifolds.
method Using the Gauss-Bonnet theorem for Riemannian simplices, the paper shows that for closed nonpositively curved 4-manifolds with nonzero Euler characteristic, simplicial volume is positive.
result The paper confirms conjectures about the relationship between simplicial volume and Euler characteristic for four-dimensional manifolds.
A notion of up and down Grover walks on simplicial complexes are proposed and their properties are investigated. These are abstract Szegedy walks, which is a special kind of unitary operators on a Hilbert space. The operators introduced in the present paper are usual Grover walks on graphs defined by using combinatoria…
Study shows range of simplicial volumes for open manifolds.
problem Understanding simplicial volumes of open manifolds.
method Analyzes locally finite simplicial volumes in dimensions at least 4.
result Set of simplicial volumes is [0, ∞] for open manifolds.
An additional minimal simplicial n-complex contains a non-splittable link in R^(2n).
problem Constructing minimal simplicial n-complexes with a specific link property.
method Presenting a new minimal simplicial n-complex.
result An additional simplicial n-complex with the same link property.
The simplicial volume of oriented closed connected smooth manifolds that admit a non-trivial smooth S1-action vanishes. In the present work we prove a version of this result for the integral foliated simplicial volume of aspherical manifolds: The integral foliated simplicial volume of aspherical oriented closed conn…
Let M be the interior of a connected, oriented, compact manifold V of dimension at least 2. If each path component of ∂V has amenable fundamental group, then we prove that the simplicial volume of M is equal to the relative simplicial volume of V and also to the geometric (Lipschitz) simplicial volume…
We establish a functor Kan from local Kan simplicial manifolds to weak Kan simplicial manifolds. It gives a solution to the problem of extending local Lie groupoids to Lie 2-groupoids.
We construct non-constructible simplicial d-spheres with d+10 vertices and non-constructible, non-realizable simplicial d-balls with d+9 vertices for d≥3.
New techniques prove bounded acyclicity results for semi-simplicial sets.
problem Homological stability of semi-simplicial sets and groups.
method Combinatorial techniques in simplicial bounded cohomology.
result Slope-1/2 stability results for certain groups. We introduce the 2-simplicial Transformer, an extension of the Transformer which includes a form of higher-dimensional attention generalising the dot-product attention, and uses this attention to update entity representations with tensor products of value vectors. We show that this architecture is a useful inductive …
This paper proposes a new method for learning covers of geometric datasets to improve topological inference and visualization.
problem Improving topological inference and visualization of large-scale geometric datasets.
method Proposes a method for learning topologically-faithful covers of geometric datasets using optimization.
result Simplicial complexes obtained from learned covers outperform standard methods in terms of size and representation of large-scale topology.
Algorithm calculates Hopf invariant for simplicial mappings.
problem Computing Hopf invariant for simplicial mappings.
method Proposed an algorithm based on Whitehead's integral formula.
result Algorithm successfully computes Hopf invariant.
The study shows simplicial volume finiteness for certain manifolds with amenable fundamental groups.
problem Determining the simplicial volume of manifolds with specific properties.
method Analyzing the fundamental group and using amenability properties.
result Simplicial volume is finite for certain manifolds with amenable fundamental groups.
The simplicial volume of non-R^3 contractible 3-manifolds is infinite.
problem Characterizing contractible 3-manifolds based on their simplicial volume.
method Analyzing the simplicial volume of contractible 3-manifolds and open 3-manifolds.
result The Euclidean space is the unique contractible 3-manifold with vanishing minimal volume.
We show that there exist closed manifolds with arbitrarily small transcendental simplicial volumes. Moreover, we exhibit an explicit uncountable family of (transcendental) real numbers that are not realised as the simplicial volume of a closed manifold.
Hypernetworks are simplified simplicial complexes with curvature.
problem Representing hypernetworks geometrically for analysis.
method Hypernetworks are interpreted as posets, which are simplicial complexes with Forman Ricci curvature.
result Hypernetworks have intrinsic curvature that correlates with their Euler characteristic.