The paper proves an infinite double bubble theorem in higher dimensions.
problem Characterizing minimizing partitions of infinite and finite volumes in Rn. method Proves a variant of the double bubble theorem for configurations with infinite and finite chambers.
result Locally minimizing (1,2)-clusters are unique in Rn for n≤7 and n≥8 under certain conditions. A lens cluster minimizes perimeter in the plane with given area constraints.
problem Minimizing perimeter in the plane with given area constraints.
method Analyzing lens clusters consisting of circular arcs with specific geometric properties.
result Lens clusters are local minimizers of the total perimeter functional.
Defines spectral selectors on lens spaces for contactomorphisms.
problem Understanding the geometry of contactomorphism groups on lens spaces.
method Using Givental's non-linear Maslov index, defines spectral selectors.
result Standard Reeb flow is a geodesic for specific lens spaces.
Clustering ensemble, or consensus clustering, has emerged as a powerful tool for improving both the robustness and the stability of results from individual clustering methods. Weighted clustering ensemble arises naturally from clustering ensemble. One of the arguments for weighted clustering ensemble is that elements (…
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.
Unified framework for DR and clustering using Gromov-Wasserstein.
problem Capturing structure in high-dimensional datasets.
method Distributional reduction framework using Gromov-Wasserstein.
result Unified approach recovers DR and clustering as special cases.
A new fuzzy k-means algorithm for high-dimensional data with variable feature weights.
problem Clustering high-dimensional data with varying feature significance.
method Proposes a modified fuzzy k-means algorithm using two entropy terms to weight features.
result Improved clustering performance on various datasets compared to state-of-the-art methods.
Proposes a weighted conformal approach for cluster label uncertainty.
problem Cluster label uncertainty in unlabeled data.
method Develops a conformal inference algorithm to correct label mismatch.
result Improves confidence set size in nonlinear and high-dimensional clustering.
New method clusters weighted directed networks using motifs.
problem Clustering directed networks fails to consider higher-order structure and edge weights.
method Motif-based weighted spectral clustering with new matrix formulae.
result Scalable and effective clustering on large graphs and real-world data.
As a model problem for clustering, we consider the densest k-disjoint-clique problem of partitioning a weighted complete graph into k disjoint subgraphs such that the sum of the densities of these subgraphs is maximized. We establish that such subgraphs can be recovered from the solution of a particular semidefinite re…
Proposes a new model for clustering multiplex networks with compositional data.
problem Clustering multiplex networks with multiple types of relations and compositional data.
method Multiplex Dirichlet stochastic block model for compositional networks.
result Validated through simulation and applied to international export data.
Exploiting different representations, or views, of the same object for better clustering has become very popular these days, which is conventionally called multi-view clustering. Generally, it is essential to measure the importance of each individual view, due to some noises, or inherent capacities in description. Many…
DMFAW improves multi-view clustering with adaptive weights and feature selection.
problem Lack of effective feature selection and empirical hyperparameter selection in existing deep matrix factorization methods.
method Introduces Deep Matrix Factorization with Adaptive Weights (DMFAW) for multi-view clustering, incorporating feature selection and dynamically updating weights using Control Theory.
result DMFAW outperforms state-of-the-art methods in clustering performance.
Proposes WM-NMF for better multi-view clustering.
problem Learning multi-view data with unequal view information content.
method Introduces a weighted multi-view NMF algorithm to learn view-specific and observation-specific weights.
result Achieves better clustering performance and handles noisy data.
New method clusters matrix-valued data by latent variables.
problem Clustering matrix-valued data with hidden structure.
method Latent variable model with hierarchical clustering.
result Algorithm attains clustering consistency in high dimensions.
Robust feature-weighted jump models for time-dependent clustering
problem Temporal clustering
method Robust feature-weighted jump model
result Accurate recovery of true cluster sequence and feature identification
We introduce a principled method for the signed clustering problem, where the goal is to partition a graph whose edge weights take both positive and negative values, such that edges within the same cluster are mostly positive, while edges spanning across clusters are mostly negative. Our method relies on a graph-based …
Locally isoperimetric partitions minimize perimeter in space.
problem Finding minimal perimeter partitions in space.
method Proving closure theorem to limit sequences of isoperimetric clusters.
result Examples of isoperimetric partitions in various dimensions.
Study of g-vector cones in cluster algebras from weighted orbifolds.
problem Determine the closure of g-vector cones in cluster algebras. method Analyzing g-vector cones in a cluster algebra defined from a weighted orbifold. result Closure of the union of g-vector cones is Rn except for specific weighted orbifolds. Improved accuracy in machine learning with Cross-Cluster Weighted Forests.
problem Improving accuracy in machine learning algorithms for datasets with clusters.
method Ensembling Random Forest learners trained on clusters determined by k-means.
result Significant improvements in accuracy and generalizability over traditional Random Forest.
The classical k-means algorithm for partitioning n points in Rd into k clusters is one of the most popular and widely spread clustering methods. The need to respect prescribed lower bounds on the cluster sizes has been observed in many scientific and business applications. In this paper, we present an…
New method embeds phylogenetic trees for clustering, recovering evolutionary relationships.
problem Lack of a meaningful way to embed phylogenetic trees into a vector space.
method Split-weight embedding to fit clustering algorithms to phylogenetic trees.
result Split-weight embedding recovers meaningful evolutionary relationships in simulated and real data.
A new model clusters network nodes based on relative edge weights.
problem Clustering networks ignores node capacities, leading to biased results.
method Proposes a Dirichlet stochastic block model for composition-weighted networks.
result Validated on simulated and real-world networks, showing improved clustering accuracy.
Proposes a method for multi-view clustering that considers local structures and feature weights.
problem Challenges in effectively exploiting complementary information across multiple views.
method Simultaneously assigns weights to different features and captures local information in view-specific feature spaces.
result Achieves state-of-the-art performance on benchmark datasets.
New clustering method reduces data redundancy for better summaries.
problem Redundancies in data summaries limit their effectiveness in large datasets.
method Khatri-Rao clustering extends centroid-based clustering to produce more succinct summaries.
result Khatri-Rao k-Means and deep clustering frameworks produce more succinct summaries with similar accuracy.
We study the use of power weighted shortest path distance functions for clustering high dimensional Euclidean data, under the assumption that the data is drawn from a collection of disjoint low dimensional manifolds. We argue, theoretically and experimentally, that this leads to higher clustering accuracy. We also pres…
Develops a new random forest method for clustered data with improved prediction and inference.
problem Improving prediction and inference accuracy for clustered data with within-cluster dependence.
method Clustered Random Forests, using weighted least squares estimators for leaf predictions.
result Optimal prediction and inference weights vary under covariate shift, necessitating user-chosen weights.
We define a class of Euclidean distances on weighted graphs, enabling to perform thermodynamic soft graph clustering. The class can be constructed form the "raw coordinates" encountered in spectral clustering, and can be extended by means of higher-dimensional embeddings (Schoenberg transformations). Geographical flow …
A new method for clustering functional data outperforms existing methods.
problem Clustering heterogeneous functional linear regression data.
method funWeightClust, a family of parsimonious models based on cluster weighted models.
result funWeightClust outperforms existing methods in simulations and real-world traffic analysis.
A family of parsimonious Gaussian cluster-weighted models is presented. This family concerns a multivariate extension to cluster-weighted modelling that can account for correlations between multivariate responses. Parsimony is attained by constraining parts of an eigen-decomposition imposed on the component covariance …
New concept of mixture complexity helps detect gradual clustering changes.
problem Determining the number of clusters in mixture models with overlaps and weight biases.
method Introducing mixture complexity (MC) as a new measure of cluster size, defined from information theory.
result MC can detect gradual clustering changes, allowing earlier detection and finer distinction.
Data clustering has received a lot of attention and numerous methods, algorithms and software packages are available. Among these techniques, parametric finite-mixture models play a central role due to their interesting mathematical properties and to the existence of maximum-likelihood estimators based on expectation-m…
Paper establishes universal lower bounds and optimal rates for clustering sub-exponential mixture models.
problem Achieving optimal error rates in clustering sub-exponential mixture models.
method Establishes universal lower bounds and demonstrates iterative algorithms' optimality in sub-exponential mixture models.
result Iterative algorithms achieve the universal lower bound in sub-exponential mixture models.
The clustering ensemble technique aims to combine multiple clusterings into a probably better and more robust clustering and has been receiving an increasing attention in recent years. There are mainly two aspects of limitations in the existing clustering ensemble approaches. Firstly, many approaches lack the ability t…
We formulate weighted graph clustering as a prediction problem: given a subset of edge weights we analyze the ability of graph clustering to predict the remaining edge weights. This formulation enables practical and theoretical comparison of different approaches to graph clustering as well as comparison of graph cluste…
We consider a decomposition method for compressive streaming data in the context of online compressive Robust Principle Component Analysis (RPCA). The proposed decomposition solves an n-ℓ1 cluster-weighted minimization to decompose a sequence of frames (or vectors), into sparse and low-rank components, from com…
clusterBMA combines clustering results from multiple models using Bayesian model averaging.
problem Uncertainty in model selection for clustering.
method Bayesian model averaging to combine results from multiple clustering algorithms.
result ClusterBMA offers probabilistic cluster allocations and quantifies model-based uncertainty.
Hierarchical clustering uses OWA operators to generalize linkage methods and avoid dendrogram inversions.
problem Avoiding unaesthetic inversions in hierarchical clustering dendrograms.
method OWA-based linkages combined with the Lance-Williams formula and conditions on weight generators.
result Conditions for weight generators to produce dendrograms without inversions.
A new metric framework for weighted projective spaces improves clustering and analysis.
problem Proximity measurement in weighted projective spaces with intrinsic scaling and topology.
method Hierarchical clustering framework based on Finsler geometry, quotienting weighted scaling action.
result The constructed metric dF satisfies the triangle inequality, making it a genuine metric. Let g be a Riemannian metric for Rd (d≥3) which differs from the Euclidean metric only in a smooth and strictly convex bounded domain M. The lens rigidity problem is concerned with recovering the metric g inside M from the corresponding lens relation on the boundary ∂M. In this paper…
Local graph clustering improves with noisy labels, enhancing accuracy and performance.
problem Local graph clustering with noisy labels for node information.
method Constructing a weighted graph with noisy labels and using diffusion-based clustering.
result Diffusion in the weighted graph yields more accurate recovery of target clusters.
A new clustering method for functional data using skewed distributions.
problem Clustering functional data with skewed distributions.
method Mixtures of functional linear regression models and three skewed multivariate distributions (variance-gamma, skew-t, normal-inverse Gaussian).
result The proposed method funWeightClustSkew performs well on simulated and real data.
New bounds for convex clustering under graph connectivity.
problem Understanding clustering performance under different graph connectivity structures.
method Random walks and concentration inequalities for random graph models.
result Improved rates of convergence for centroid recovery.
Machine learning uncovers hidden patterns in Calabi-Yau hypersurfaces.
problem Identifying and clustering Calabi-Yau hypersurfaces from weighted-P4s.
method Supervised and unsupervised machine learning techniques.
result High accuracy in predicting topological parameters and identifying hypersurfaces.
Clustering is one of the major roles in data mining that is widely application in pattern recognition and image segmentation. Fuzzy C-means (FCM) is the most used clustering algorithm that proven efficient, fast and easy to implement, however, FCM uses the Euclidean distance that often leads to clustering errors, espec…
In this paper we analyze the local and global boundary rigidity problem for general Riemannian manifolds with boundary (M,g). We show that the boundary distance function, i.e., dg∣∂M×∂M, known near a point p∈∂M at which ∂M is strictly convex, determines g in a suita…
Improved spectral clustering for community detection in networks.
problem Community detection in networks.
method Improved spectral clustering (ISC) based on k-means clustering on weighted eigenvectors of a regularized Laplacian matrix.
result ISC yields stable consistent community detection under mild conditions and outperforms classical methods.
In mixture model-based clustering applications, it is common to fit several models from a family and report clustering results from only the `best' one. In such circumstances, selection of this best model is achieved using a model selection criterion, most often the Bayesian information criterion. Rather than throw awa…