Geodesic clustering improves latent space clustering in deep generative models.
problem Latent representations in deep generative models distort semantic distances, making clustering difficult.
method Proposed an efficient algorithm for computing geodesics and distances in the latent space, accounting for its distortion.
result Geodesic distance reflects the internal structure of the data, improving clustering performance.
New clustering method for uncertain data using Wasserstein barycenters.
problem Clustering uncertain and structured data with observational/experimental error.
method Wasserstein barycenters and geodesic criterion for optimal clustering.
result Effective clustering of complex data in astronomy, biology, and remote sensing.
This paper advocates a novel framework for segmenting a dataset in a Riemannian manifold M into clusters lying around low-dimensional submanifolds of M. Important examples of M, for which the proposed clustering algorithm is computationally efficient, are the sphere, the set of positive definite matrices, and the…
Proves existence of many non-R-covered Anosov flows on hyperbolic 3-manifolds.
problem Existence of many non-R-covered Anosov flows on hyperbolic 3-manifolds. method Description of clusters of lozenges in orbit spaces of constructed Anosov flows.
result Existence of hyperbolic 3-manifolds carrying many pairwise orbitally inequivalent quasi-geodesic Anosov flows.
New quasi-geodesics for Stiefel manifold simplify complex computations.
problem Efficiently solving geodesic endpoint problem on Stiefel manifold.
method Derived new representations of quasi-geodesics for large-scale computations.
result New quasi-geodesics are closer to Riemannian geodesics.
New method estimates geodesic distances using spherelets.
problem Accurately estimating geodesic distances on unknown manifolds.
method Uses spherelets to locally approximate unknown subspaces and estimate geodesic distances.
result Lower error for many manifolds, validated through simulations and real data.
We present a methodology for clustering N objects which are described by multivariate time series, i.e. several sequences of real-valued random variables. This clustering methodology leverages copulas which are distributions encoding the dependence structure between several random variables. To take fully into account …
Efficient algorithm for self-directed learning of convex clusters on graphs.
problem Self-directed classification of nodes on graphs with convex clusters.
method Developed efficient algorithms for (geodesically) convex clusters on graphs.
result Polynomial runtime algorithm with 3(h(G)+1)4lnn mistakes for graphs with two convex clusters. Using geodesic currents, we provide a theoretical justification for some of the experimental results regarding the behavior of Whitehead's algorithm on non-minimal inputs, that were obtained by Haralick, Miasnikov and Myasnikov via pattern recognition methods. In particular we prove that the images of "random" elements…
Establishes a link between heat diffusion and manifold distances in data.
problem No theoretical link between diffusion-based manifold learning and geodesic distances.
method Formulates heat geodesic embeddings based on Riemannian geometry.
result Method outperforms state-of-the-art in preserving manifold distances and cluster structure.
The Dehornoy order on braid groups is derived from a cluster algebra.
problem Deriving the Dehornoy order on braid groups from a cluster algebra.
method Using a tracial state on a cluster C∗-algebra associated to a surface. result The Dehornoy order on B2g+n is recovered from the cluster algebra. In this paper, we are interested in the location of conjugate points along a geodesic in the volumorphism group of a compact three-dimensional manifold without boundary (the configuration space of an ideal fluid). As shown in the author's previous work, these are typically pathological, i.e., they can occur in clusters…
For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller space of the surface. On the geometric side, this requires opening the surface at each interior marked point into an addit…
Unified proof of Aigner's conjectures using geodesics.
problem Proving conjectures related to Markov numbers.
method Using geodesics on the punctured torus.
result Unified proof of Aigner's conjectures.
Paper introduces MPPGA for integrating multiple PGA models on Riemannian manifolds.
problem Challenges in dimensionality reduction on Riemannian manifolds with multiple modalities.
method Develops a mixture probabilistic principal geodesic analysis (MPPGA) model.
result Demonstrates improved clustering and shape analysis using MPPGA.
Quantum dynamics algorithm learns manifold from data.
problem Learning manifolds from high-dimensional datasets.
method Simulation of quantum dynamics on a graph embedding of data.
result Algorithm reveals connections between data sampling and quantization.
New distances for comparing multivariate normal distributions.
problem Comparing multivariate normal distributions efficiently and accurately.
method Approximated Fisher-Rao distance and pullback SPD cone distances.
result Efficient computation of distances between normal distributions.
Graph regularized autoencoder improves anomaly detection performance.
problem Unsupervised anomaly detection in high-dimensional data.
method Developed a graph regularized autoencoder using MST-based distances.
result Outperforms alternative methods on 20 benchmark anomaly detection datasets.
We analyze convergence of Fermat distances and their application in clustering.
problem Understanding convergence properties of Fermat distances on Riemannian manifolds.
method Geometric and statistical arguments in percolation theory, leveraging novel arguments for non-uniform densities and curved domains.
result Discrete, sample-based Fermat distances converge to their continuum analogues with a precise rate dependent on intrinsic dimensionality.
Proposes a fuzzy rule-based method for data visualization.
problem Preserving neighborhood relationships and handling non-linear manifolds in data visualization.
method Uses a first-order Takagi-Sugeno model with clusters and Geodesic c-means clustering for rule generation and parameter estimation.
result Behaves desirably and performs better than or comparable to other methods.
In this paper, we present GASG21 (Grassmannian Adaptive Stochastic Gradient for L2,1 norm minimization), an adaptive stochastic gradient algorithm to robustly recover the low-rank subspace from a large matrix. In the presence of column outliers, we reformulate the batch mode matrix L2,1 norm minimization with…
In this article, we will formulate a mathematical framework that allows us to treat character animations as points on infinite dimensional Hilbert manifolds. Constructing geodesic paths between animations on those manifolds allows us to derive a distance function to measure similarities of different motions. This appro…
New framework tracks communities in dynamic networks.
problem Discovering and tracking communities in evolving networks.
method Spectral framework on Grassmann manifold for subspace tracking.
result Improved dynamic community detection results across various network types.
New method recovers manifold distances from noisy data.
problem Reconstructing manifold geometry from noisy distance measurements.
method Develops new framework to estimate L2-norms of expectation-functions, uses geometric clusters to recover distances.
result Recovery of true distances up to an additive error of O(ε log ε⁻¹) under mild geometric assumptions.
We adapt the method of Simon [JDG '93] to prove a C1,α-regularity theorem for minimal varifolds which resemble a cone C02 over an equiangular geodesic net. For varifold classes admitting a "no-hole" condition on the singular set, we additionally establish C1,α-regularity near the cone $\bf{C}_0^2 \ti…
Bregman divergences play a central role in the design and analysis of a range of machine learning algorithms. This paper explores the use of Bregman divergences to establish reductions between such algorithms and their analyses. We present a new scaled isodistortion theorem involving Bregman divergences (scaled Bregman…
We relax indicator matrices to form a manifold for faster optimization.
problem Optimizing indicator matrices is NP-hard.
method Developed a Riemannian manifold (RIM) and Riemannian optimization methods.
result RIM manifold optimization is significantly faster and yields better results.
The trace set of a Fuchsian group Γ ist the set of length of closed geodesics in the surface Γ\H. Luo and Sarnak showed that the trace set of a cofinite arithmetic Fuchsian group satisfies the bounded clustering property. Sarnak then conjectured that the B-C property actually characterizes arithm…
Statistical shape analysis can be done in a Riemannian framework by endowing the set of shapes with a Riemannian metric. Sobolev metrics of order two and higher on shape spaces of parametrized or unparametrized curves have several desirable properties not present in lower order metrics, but their discretization is stil…
A new growth model for dynamic networks using Markovian latent points.
problem Modeling temporal dynamic networks with latent points and distances.
method Markovian latent space dynamic with Euclidean Sphere sampling and connection probabilities based on geodesic distances.
result Theoretical guarantees for non-parametric estimation of the latitude and envelope functions.
Convex clustering can only learn convex clusters, with significant gaps between clusters.
problem Understanding the limitations and capabilities of convex clustering.
method Analyzing convex clustering solutions, proving properties, and characterizing clusters.
result Convex clustering can only learn convex clusters with significant gaps between clusters.
Proposes a method to predict cluster number and cluster representatives using cluster stability analysis.
problem Determining the number of clusters in a dataset.
method Analyzes cluster stability using Monte-Carlo simulation to predict cluster number and find cluster representatives.
result Significant improvement in predicting cluster numbers and cluster composition in large datasets.
Paper compares clusterability measures for data mining.
problem Selecting appropriate clusterability measures for data.
method Extensive comparison of clusterability measures.
result Guidelines for selecting suitable measures for clustering.
Proposes a new clustering method based on expectiles for non-spherical clusters.
problem Inability of K-means to handle non-spherical clusters. method Uses expectiles to define cluster centers and searches for clusters via a greedy algorithm.
result Outperforms K-means and spectral clustering on asymmetric shaped clusters. CCMM efficiently solves large-scale convex clustering problems.
problem Scalability and hierarchical structure in convex clustering.
method Majorization-minimization algorithm with cluster fusions and efficient updating.
result CCMM achieves efficient solutions for large datasets.
This paper reviews weighted clustering ensemble methods.
problem Improving clustering results from individual methods.
method Different types of weights and approaches to determining weight values.
result Unified framework for selecting appropriate weighting mechanisms.
Round spheres are uniquely characterized by half-geodesics.
problem Characterizing round spheres in Riemannian geometry.
method Establishing that Riemannian spheres with specific geodesic properties are round.
result Riemannian spheres with all geodesics closed and many half-geodesics are round.
Discussing issues in robust clustering, especially with Gaussian models.
problem Handling outliers and ambiguity in clustering groups.
method Focus on Gaussian mixture model, examining formal definitions, interactions, and tuning decisions.
result Outliers can confuse clustering groups and existing stability measures fail with them.
New indices for determining cluster compactness and separability.
problem Challenges in identifying true clusters in data sets.
method Developed absolute cluster indices to measure compactness and separability.
result Demonstrated improved performance compared to existing indices.
Mode clustering is a nonparametric method for clustering that defines clusters using the basins of attraction of a density estimator's modes. We provide several enhancements to mode clustering: (i) a soft variant of cluster assignment, (ii) a measure of connectivity between clusters, (iii) a technique for choosing the …
In this paper, a similarity-driven cluster merging method is proposed for unsuper-vised fuzzy clustering. The cluster merging method is used to resolve the problem of cluster validation. Starting with an overspecified number of clusters in the data, pairs of similar clusters are merged based on the proposed similarity-…
This paper introduces a persistence metric to compare clustering solutions with different numbers of clusters.
problem Determining the true number of clusters in a dataset when prior knowledge is lacking.
method The paper introduces a persistence metric based on the maximum over two-norms of all cluster-covariance matrices.
result The persistence metric accurately identifies clustering solutions with the true number of clusters.
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…
Study examines how cluster number affects short-text clustering, introducing a stability metric.
problem Challenges in finding meaningful clusters in short-text data.
method Introduces a stability metric to determine cluster robustness and visualizes cluster subdivisions.
result Choosing a cluster number involves balancing informativeness and complexity, not seeking a single 'optimal' solution.
Convex clustering, a convex relaxation of k-means clustering and hierarchical clustering, has drawn recent attentions since it nicely addresses the instability issue of traditional nonconvex clustering methods. Although its computational and statistical properties have been recently studied, the performance of convex c…
Cluster LOCO: A model-agnostic feature importance score for interpreting cluster outputs
problem Interpreting and auditing cluster outputs
method Cluster LOCO (Leave-One-Covariate-Out)
result More reliably recovers informative features than existing methods
Clustering is a central approach for unsupervised learning. After clustering is applied, the most fundamental analysis is to quantitatively compare clusterings. Such comparisons are crucial for the evaluation of clustering methods as well as other tasks such as consensus clustering. It is often argued that, in order to…
Proposes a deep density-based image clustering method.
problem Challenges in clustering images with unknown cluster number and shape.
method Two-stage approach: feature extraction with CAE and t-SNE, followed by density-based clustering.
result Achieves clustering performance comparable to state-of-the-art methods.