Study proves topological complexity and LS-category inequalities for specific groups and manifolds.
problem Proving inequalities for topological complexity and LS-category of specific groups and manifolds.
method Analyzing torsion free hyperbolic and nilpotent groups, lens spaces, using inequalities and counter-examples.
result Proves inequalities for topological complexity and LS-category of specific groups and manifolds.
New probabilistic invariants bound classical topological complexity and category.
problem Bounding classical topological complexity and category.
method Developed probabilistic variants of one-category and diagonal topological complexity.
result Identified new invariants with distributional category and complexity on Eilenberg-Mac Lane spaces.
Defines new versions of distributional topological complexity for spaces.
problem Generalizing topological complexity to sequences.
method Introduces a sequence of higher versions of distributional topological complexity.
result The new versions are homotopy invariants and relate to distributional LS-category.
Piecewise normalizing flows improve multi-modal distribution modeling.
problem Improving accuracy in modeling multi-modal distributions.
method Divide target distribution into clusters, train flows to match standard normal base.
result Piecewise flows outperform standard approaches in accuracy.
New TC variant dTC better fits motion planning for some systems.
problem Improving motion planning for autonomous systems.
method Defined and computed new homotopy invariant dTC.
result dTC and dcat provide better motion planning solutions.
This paper provides new algorithms for distributed clustering for two popular center-based objectives, k-median and k-means. These algorithms have provable guarantees and improve communication complexity over existing approaches. Following a classic approach in clustering by \cite{har2004coresets}, we reduce the proble…
Proposes TSBP for matching topological signal distributions.
problem Matching signal distributions on topological domains.
method Topological Schrödinger Bridge (TSBP) with linear topology-aware stochastic dynamics.
result Derives closed-form topological SB (TSB) for Gaussian boundary distributions.
We study a model of wealth dynamics [Bouchaud and Mézard 2000, \emph{Physica A} \textbf{282}, 536] which mimics transactions among economic agents. The outcomes of the model are shown to depend strongly on the topological properties of the underlying transaction network. The extreme cases of a fully connected and a ful…
A method for learning distributions on complex manifolds using normalizing flows.
problem Learning distributions on non-Euclidean manifolds with high efficiency and accuracy.
method Learning a distribution on a manifold by combining local models that form an open cover.
result The method achieves better sample efficiency and competitive performance on manifolds of unknown topology.
CMRFs extend PGMs for topological data, capturing both conditional and marginal dependencies.
problem Limited expressiveness of PGMs for topological data.
method Introducing Colored Markov Random Fields (CMRFs) that model Gaussian edge variables on topological spaces.
result CMRFs improve distributed estimation over physical networks compared to baselines.
Power system studies require the topological structures of real-world power networks; however, such data is confidential due to important security concerns. Thus, power grid synthesis (PGS), i.e., creating realistic power grids that imitate actual power networks, has gained significant attention. In this letter, we cas…
We study the wealth distribution of the Bouchaud--Mézard (BM) model on complex networks. It has been known that this distribution depends on the topology of network by numerical simulations, however, no one have succeeded to explain it. Using "adiabatic" and "independent" assumptions along with the central-limit theore…
This paper examines ADMM for network averaging, revealing its convergence rates and network topology impacts.
problem Efficiently averaging local information over a network using ADMM.
method Comparative analysis of ADMM and other algorithms on a canonical distributed averaging problem.
result Characterization of ADMM convergence and optimal parameter tuning based on network spectral properties.
Large datasets represented by multidimensional data point clouds often possess non-trivial distributions with branching trajectories and excluded regions, with the recent single-cell transcriptomic studies of developing embryo being notable examples. Reducing the complexity and producing compact and interpretable repre…
Study uses topological signatures to quantify financial market complexity.
problem Capturing temporal organization beyond volatility measures.
method Null validated topological approach using L1 norm of persistence landscapes. result Persistence landscape norms reveal dynamical structure during market stress.
Strong regulations in the financial industry mean that any decisions based on machine learning need to be explained. This precludes the use of powerful supervised techniques such as neural networks. In this study we propose a new unsupervised and semi-supervised technique known as the topological hierarchical decomposi…
Study numerical invariants for groups, computing for cyclic groups and surfaces.
problem Numerical invariants for groups and their computation.
method Computational and theoretical analysis of groups, including finite cyclic groups and nonorientable surfaces.
result Formula for the numerical invariant of free products of groups.
Proves conjecture on graph configuration spaces' complexity.
problem Topological complexity of graph configuration spaces.
method Lower bound derived from insights into aspherical spaces.
result Proves Farber's conjecture on stable topological complexity.
New calculations of topological complexity for symplectic CW-complexes.
problem Calculating topological complexity for symplectic CW-complexes.
method Using atoroidal cohomology classes and CW-complexes, proving topological complexity for symplectic spaces.
result Every atoroidally symplectic CW-complex of dimension 2n has topological complexity 4n.
A new base distribution for normalizing flows allows modeling complex distributions without sacrificing invertibility.
problem Normalizing flows struggle with complex, non-trivial distributions.
method Learned rejection sampling for base distribution, combined with optimization of log-likelihood and Kullback-Leibler divergence.
result The method effectively models complicated distributions without sacrificing invertibility.
The paper explores conditions for topological rigidity in quotients of the Davis complex.
problem Understanding when quotients of the Davis complex are topologically rigid.
method Analyzing quotients of the Davis complex of right-angled Coxeter groups and conditions on defining graphs.
result Introduction of infinitely many infinite topologically rigid subclasses.
Paper uses TDA to assess cryptocurrency risk by measuring phase space instability.
problem Traditional risk measures fail to capture market dynamics' geometric structure.
method Applied Takens' Delay Embedding Theorem to generate point cloud, computed persistent homology groups, defined Topological Persistence Norm.
result Proposed leverage calibration heuristic based on persistence of 1-dimensional cycles.
The paper develops algorithms and topological invariants for distinguishing dynamic systems.
problem Distinguishing the topological type of surfaces and functions in dynamic systems.
method Construction of algorithms and topological invariants using discrete topological structures.
result The development of discrete topological structures for topological equivalence of dynamic systems.
The article applies Lusternik-Schnirelmann theory to establish lower bounds on critical points using sequential and parametrized topological complexity.
problem Establishing lower bounds on the number of critical points of functions using topological complexity.
method Applying Lusternik-Schnirelmann theory to sequential and parametrized topological complexity.
result Established various lower bounds on the number of critical points using sequential and parametrized topological complexity.
A new approach uses circuit topology to study complex polymer interactions.
problem Understanding structural phase transitions in entangled polymer systems.
method Braided circuit topology framework for multiple-chain systems.
result Circuit topological motif fractions are effective order parameters for structural transitions.
Constructs algorithms to recognize and classify 2D surfaces.
problem Recognizing and classifying 2D surfaces in dynamic systems.
method Discrete topological structures and algorithms for simplicial and CW-complexes.
result Determines the topological type of 2-manifolds.
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.
We introduce a Bayesian approach to discovering patterns in structurally complex processes. The proposed method of Bayesian Structural Inference (BSI) relies on a set of candidate unifilar HMM (uHMM) topologies for inference of process structure from a data series. We employ a recently developed exact enumeration of to…
We present a new approach to equivariant version of the topological complexity, called a symmetric topological complexity. It seems that the presented approach is more adequate for the analysis of an impact of symmetry on the the motion planning algoritm than the one introduced and studied by Colman and Grant. We show …
CT improves neural network performance on cell complex data.
problem Improving predictive performance of neural networks on complex data.
method Introducing the Cellular Transformer (CT) that generalizes graph-based transformers to cell complexes.
result CT achieves state-of-the-art performance on cell complex datasets without complex enhancements.
We present some results supporting the Iwase-Sakai conjecture about coincidence of the topological complexity TC(X) and monoidal topological complexity TCM(X). Using these results we provide lower and upper bounds for the topological complexity of the wedge X∨Y. We use these bounds to give a counterexample t…
We provide an upper bound on the topological complexity of twisted products. We use it to give an estimate TC(X)≤TC(π1(X))+dimX of the topological complexity of a space in terms of its dimension and the complexity of its fundamental group.
Study topological invariants of complexes for Riemannian manifolds.
problem Understanding topological properties of Riemannian manifolds.
method Analyzing Betti numbers and Euler characteristic of Vietoris-Rips and Čech complexes.
result Betti curve converges to manifold's Betti number within a scale parameter interval.
Topological complexity for closed 1-forms
problem Topological complexity for closed 1-forms
method Introduce and study a corresponding version of topological complexity
result Establish analogues of basic properties of ordinary topological complexity
Lecture notes on curves in complex projective plane from a topological viewpoint.
problem Understanding curves in complex projective plane from a topological perspective.
method Topological analysis of curves in complex projective plane.
result Curves in complex projective plane have unique topological properties.
Unified framework for measuring concentration in weighted networks considering both weight distributions and network structure.
problem Traditional indices neglect the topology of relationships among network elements.
method Develops a family of topology-aware concentration indices that jointly account for weight distributions and network structure.
result The proposed indices preserve key properties and allow concentration to be evaluated across different dimensions of dependence.
We expand Topological Field Theory on some special CW-complexes (brane complexes). This Brane Topological Field Theory one-to-one corresponds to infinite dimensional Frobenius Algebras, graduated by CW-complexes of lesser dimension. We define general and regular Hurwitz numbers of brane complexes and prove that they ge…
The increasing penetration of distributed energy resources poses numerous reliability issues to the urban distribution grid. The topology estimation is a critical step to ensure the robustness of distribution grid operation. However, the bus connectivity and grid topology estimation are usually hard in distribution gri…
Proves planar graphs' configuration spaces have highest topological complexity.
problem Proving Farber's conjecture for planar graphs.
method Generic maximality argument for topological complexities.
result Generic maximality of topological complexities for planar graphs.
In this paper we study a notion of topological complexity for the motion planning problem. The topological complexity is a number which measures discontinuity of the process of motion planning in the configuration space X. More precisely, it is the minimal number k such that there are k different motion planning rules,…
Feature maps, that preserve the global topology of arbitrary datasets, can be formed by self-organizing competing agents. So far, it has been presumed that global interaction of agents is necessary for this process. We establish that this is not the case, and that global topology can be uncovered through strictly local…
Graph braid groups' complexity stabilizes for most graphs.
problem Stabilization of topological complexity in graph braid groups.
method Geometric lower bounds on configuration spaces.
result Topological complexity stabilizes for most graphs.
Study numerical invariants under retraction maps between topological spaces.
problem Understand behavior of invariants like cohomological dimensions under retractions.
method Introduced a notion of retraction and studied several numerical invariants.
result Proved inequalities between invariants hold under retractions.
Studies amenable category's monotonicity and its relation to topological complexity.
problem Monotonicity of amenable category for degree-one maps.
method Uses amenable covers and compares with topological complexity.
result Establishes a relation between amenable category and topological complexity.
Survey on the topology of singular foliations in complex 2-space.
problem Understanding the topology of singular foliations in complex 2-space.
method Overview and survey of existing research.
result Overview of current knowledge on foliation singularities.
The paper develops a scalable method to infer GRNs from sparse data.
problem Inferring complex gene regulatory networks from limited and temporally sparse data.
method Bayesian optimization and kernel-based methods to construct a Gaussian Process (GP) model.
result The method efficiently searches for the topology with the highest likelihood value.
Extends HMM to topological spaces for modeling complex data.
problem Modeling complex, continuous data in infinite-dimensional spaces.
method Use of Onsager-Machlup functional and Cameron-Martin space.
result Demonstrates versatility in identifying sleep states and snowfall patterns.
Study of universal complexes in toric topology with applications in category theory.
problem Properties and applications of universal complexes in toric topology.
method Combinatorial and topological analysis of X(Fpn) and K(Fpn). result Lusternick-Schnirelmann categories of moment angle complexes calculated for universal complexes.