Study character varieties of surfaces using cluster algebras and Poisson structures.
problem Character varieties of surfaces and their Poisson structures.
method Use Bonahon-Wong's trace map and cluster algebras associated with ideal triangulations.
result Recover Goldman Poisson algebra from cluster algebra structure and show automorphisms.
This thesis studies cluster varieties from surfaces and their properties.
problem Understanding the geometry of cluster varieties associated to surfaces.
method Developed properties of symplectic double and defined measured laminations.
result Proved Fock and Goncharov's duality conjectures for quantum cluster varieties.
Quantizes canonical bases for cluster varieties of type A.
problem Deforming algebra of functions on cluster varieties.
method Natural q-deformation of Fock and Goncharov's canonical basis. result Extension to quantum symplectic double.
Study of Legendrian links using Floer theory and cluster varieties.
problem Understanding exact Lagrangian fillings of positive braid Legendrian links.
method Floer-theoretic approach and exact Lagrangian cobordisms.
result Proves that positive braid Legendrian links admit infinitely many exact Lagrangian fillings.
The paper connects Legendrian links to cluster theory and exact Lagrangian fillings.
problem Understanding the relationship between Legendrian links and cluster theory.
method Using exact Lagrangian fillings and cluster theory, the paper establishes connections between Legendrian links and cluster varieties.
result The augmentation variety of certain Legendrian 2-bridge links is isomorphic to a product of cluster varieties.
Solves Riemann-Hilbert problems on surface triangulations.
problem Riemann-Hilbert problems in Donaldson-Thomas theory.
method Map from stability conditions to cluster variety.
result Constructs solutions to Riemann-Hilbert problems.
A positive space is a space with a positive atlas, i.e. a collection of rational coordinate systems with subtraction free transition functions. The set of positive real points of a positive space is well defined. We define a tropical compactification of the latter. We show that it generalizes the Thurston compactificat…
Cluster structures on moduli spaces are linked to Legendrian knots.
problem Constructing cluster structures on moduli spaces.
method Quantization of exact Lagrangian fillings corresponding to Legendrian knots.
result Cluster structures can be deduced from Legendrian knots.
Quantum Teichmüller theory constants confirmed for cluster varieties.
problem Verifying constants in quantum Teichmüller theory representations.
method Computation of constants using quantum dilogarithm.
result All constants are confirmed to be 1, confirming genuine representations.
DeepPlace learns to place applications in clusters using RL.
problem Manual placement rules for scheduling are non-trivial and suboptimal.
method Uses Deep Reinforcement Learning to learn optimal placement rules.
result Reduces resource competition and optimizes cluster utilization.
We define a canonical map from a certain space of laminations on a punctured surface into the quantized algebra of functions on a cluster variety. We show that this map satisfies a number of special properties conjectured by Fock and Goncharov. Our construction is based on the "quantum trace" map introduced by Bonahon …
Cluster structures found on Teichmüller spaces for classical groups.
problem Understanding the structure of Teichmüller spaces for classical groups.
method Showed that spaces of framed local systems have cluster variety structures.
result Spaces of framed local systems form cluster varieties and a cluster ensemble.
The paper addresses statistical inference for cluster trees, providing methods to quantify uncertainty.
problem Quantifying uncertainty in cluster tree estimates.
method Developed metrics for comparing cluster trees, proposed methods for constructing and summarizing confidence sets, introduced a partial ordering for pruning insignificant features.
result Proposed methods for constructing and summarizing confidence sets for the unknown true cluster tree.
The paper proposes a parallelizable clustering method for multivariate data.
problem The standard model-based clustering method assumes the same number of clusters per margin, which is often unrealistic.
method Developed a finite mixture model per margin with different numbers of clusters, and used a game-inspired algorithm to cluster multivariate data.
result The proposed method shows good performance in various scenarios and real datasets.
Research connects geometric structures to knot theory and algebraic combinatorics.
problem Understanding the mixed Hodge structure on cohomology of open positroid varieties.
method Relates mixed Hodge structure to Khovanov-Rozansky homology of associated links.
result Rational q,t-Catalan numbers are derived from mixed Hodge polynomials of open positroid varieties. Study connects Morse theory with cluster variables for wall-crossing in Cerf diagrams.
problem Understanding wall-crossing in Cerf theory.
method Relates Bruhat numbers in real Morse theory to cluster variables in braid varieties.
result Provides wall-crossing coordinates in Cerf diagrams.
Quantum Teichmüller theory constants are shown to be 1.
problem Verifying genuine representations in quantum Teichmüller theory.
method Quantum dilogarithm function and unitary intertwiners.
result Algebraic relations among quantum mutations are satisfied by intertwiners with constants equal to 1.
The paper constructs bases for cluster varieties using mSL3-webs and laminations.
problem Cluster varieties associated to mSL3-local systems on surfaces. method Introducing mSL3-laminations, developing quantum and classical trace maps, and constructing bases. result Bases of regular functions on mPGL3 cluster varieties constructed from mSL3-laminations. Bracelets and theta bases match in various cluster algebras.
problem Matching bracelet and theta bases in cluster algebras.
method Comparing skein and cluster algebras, defining quantum bracelets, and analyzing cluster scattering diagrams.
result Quantum bracelets coincide with theta functions in various cluster algebras.
Proves conjecture linking cluster algebras and skein algebras for surfaces with punctures.
problem Cluster and skein algebras on surfaces with punctures.
method Geometric and algebraic methods, including decorated Teichmüller spaces and skein algebras.
result Cluster and skein algebras coincide for surfaces with at least 2 punctures.
PEA improves PCA and k-means for non-linear data and complex clusters.
problem Non-linear dimensionality reduction and clustering challenges.
method Principal Elliptical Analysis (PEA) for efficient non-linear approximation.
result PEA outperforms k-means in complex data clustering.
Algorithm for constructing cluster structures on moduli spaces for general reductive groups.
problem Constructing cluster structures on moduli spaces for general reductive groups.
method Algorithm for constructing cluster structures based on mild hypotheses.
result Cluster structure constructed for groups of type G2, and illustrated for type A.
The paper connects knot theory and cluster algebras via dimer face polynomials.
problem Understanding the relationship between knot theory and cluster algebras.
method Analyzing dimer face polynomials and their connections to Alexander polynomials and cluster algebras.
result Dimer face polynomials are multivariate generalizations of Alexander polynomials and F-polynomials in cluster algebras. Paper analyzes conditions for clustering BMMs with unknown clusters.
problem Clustering Bernoulli Mixture Models (BMMs) with unknown number of clusters.
method Theoretical analysis of sample complexity and dimensionality for PAC-clusterability.
result First non-asymptotic bounds on sample complexity for learning or clustering BMMs.
In many practical applications of clustering, the objects to be clustered evolve over time, and a clustering result is desired at each time step. In such applications, evolutionary clustering typically outperforms traditional static clustering by producing clustering results that reflect long-term trends while being ro…
New method for matrix completion using algebraic varieties.
problem Matrix completion with high-rank data.
method Algebraic variety mapping and kernel trick.
result Algorithm recovers data up to sampling complexity bounds.
New method fills cluster seeds with exact Lagrangian structures.
problem Surjectivity of map from Lagrangian fillings to cluster seeds.
method Construction of quiver with potential and Lagrangian disk surgeries.
result Trivial deformation space of quiver allows manipulation of fillings.
In this paper, we propose a model-based clustering method (TVClust) that robustly incorporates noisy side information as soft-constraints and aims to seek a consensus between side information and the observed data. Our method is based on a nonparametric Bayesian hierarchical model that combines the probabilistic model …
New visual quality index for fuzzy clustering.
problem No accurate quality index for fuzzy clustering across datasets.
method Proposes a new visual quality index and graph-based solution.
result Validated through extensive experiments on various datasets.
We perform a large-scale analysis of language diatopic variation using geotagged microblogging datasets. By collecting all Twitter messages written in Spanish over more than two years, we build a corpus from which a carefully selected list of concepts allows us to characterize Spanish varieties on a global scale. A clu…
New algorithm for fuzzy clustering improves clustering quality.
problem Seeding iterative fuzzy algorithms for high-quality clustering.
method MaxMin Linear initialization algorithm for fuzzy C-Means.
result Validation through experiments on various datasets.
Geometric model of unbounded sl3 laminations with tropical coordinates.
problem Modeling unbounded laminations in cluster varieties.
method Introducing tropical cluster coordinates and geometric gluing procedures.
result Established a geometric gluing procedure for unbounded sl3 laminations.
The paper generalizes Thurston's earthquake map to cluster algebras of finite type.
problem Tackling Thurston's earthquake map in the context of cluster algebras of finite type.
method Introducing a cluster algebraic generalization of Thurston's earthquake map, defined by gluing exponential maps.
result Proves an analogue of the earthquake theorem for cluster algebras of finite type, showing the cluster earthquake map is a homeomorphism.
BalLOT uses optimal transport for balanced k-means clustering.
problem Balanced k-means clustering of data. method BalLOT is an optimal transport approach to alternating minimization.
result BalLOT provides theoretical guarantees for exact and partial recoveries of planted clusters.
Cluster analysis methods seek to partition a data set into homogeneous subgroups. It is useful in a wide variety of applications, including document processing and modern genetics. Conventional clustering methods are unsupervised, meaning that there is no outcome variable nor is anything known about the relationship be…
New method clusters hypergraphs using weighted random walks and Laplacians.
problem Clustering hypergraph data with edge-dependent weights.
method Random walks with edge-dependent vertex weights, constructing hypergraph Laplacians for clustering.
result Proposed methods outperform existing hypergraph clustering algorithms.
Quantizes moduli space of 3D gravity metrics.
problem Quantize moduli space of 3D gravity metrics.
method Develops geometrically natural classes of observables and uses cluster X-varieties. result Obtains projective unitary representations of mapping class group.
Paper uses GMM with DVAE to detect star clusters in noisy images.
problem Detecting stellar clusters in astronomical images.
method Unsupervised approach using Deep Variational Autoencoder (DVAE) combined with Gaussian Mixture Model (GMM).
result Method outperforms state-of-the-art in recognizing star clusters, even in noisy images.
Paper proposes a method to estimate cluster number without prior knowledge.
problem Community detection in unlabeled networks.
method Semi-definite relaxations for estimating cluster number and matrix.
result Method recovers cluster number and matrix exactly with high probability.
We describe a probabilistic (generative) view of affinity matrices along with inference algorithms for a subclass of problems associated with data clustering. This probabilistic view is helpful in understanding different models and algorithms that are based on affinity functions OF the data. IN particular, we show how(…
Paper introduces scalable clustering for large datasets with outliers.
problem Lack of scalable algorithms for large datasets with outliers.
method Provable robust clustering algorithm based on loss minimization for Gaussian mixture models.
result Algorithm provides high accuracy with theoretical guarantees and outperforms existing methods.
Study of quantum decorated character stacks and their quantizations.
problem Quantization of decorated character stacks and their compatibility with cutting and gluing.
method Using stratified factorization homology, extend Fock and Goncharov's construction to include stacky points.
result Construction of categorical charts and flips on quantum decorated character stacks.
Sparse Convex Clustering improves clustering performance in high-dimensional data.
problem Distortion in convex clustering performance with uninformative features.
method Introduces Sparse Convex Clustering with an adaptive group-lasso penalty and a tuning criterion based on clustering stability.
result Demonstrates improved clustering performance through feature selection.
End-to-end clustering without labels using normalized cuts.
problem Clustering unlabeled data without supervision.
method Learning to optimize expected normalized cuts, differentiable loss function.
result State-of-the-art results on various benchmarks, generalizing to new datasets.
Bayesian distance clustering improves robustness to kernel choice.
problem Kernel sensitivity in model-based clustering.
method Modeling pairwise distances instead of original data.
result Dramatic gains in cluster inference robustness.
Unsupervised clustering of curves according to their shapes is an important problem with broad scientific applications. The existing model-based clustering techniques either rely on simple probability models (e.g., Gaussian) that are not generally valid for shape analysis or assume the number of clusters. We develop an…
Develops a framework for consistent clustering algorithm benchmarking.
problem Lack of consistent benchmarking and diverse datasets for clustering algorithms.
method Consistent methodology, aggregated and polished datasets, interactive explorer.
result Introduces a comprehensive framework for testing clustering algorithms.
Characterizes pseudo-Anosov mapping classes on general marked surfaces.
problem Stability of mapping classes on marked surfaces.
method Cluster algebraic description and reduction procedure of mapping classes.
result Characterizes pseudo-Anosov mapping classes in terms of uniform sign stability.