New method clusters non-spherical Gaussian mixtures with fewer samples and time.
problem Clustering non-spherical Gaussian mixtures with arbitrary component covariances.
method Sum-of-Squares method for finding low-dimensional projections.
result Improved clustering algorithms with fewer samples and time complexity.
We consider the problem of clustering data points in high dimensions, i.e. when the number of data points may be much smaller than the number of dimensions. Specifically, we consider a Gaussian mixture model (GMM) with non-spherical Gaussian components, where the clusters are distinguished by only a few relevant dimens…
New methods improve clustering accuracy in noisy data sets.
problem Improving clustering accuracy in data sets with noise features.
method Feature rescaling factors to enhance clustering validity indexes.
result Our methods increase the likelihood of estimating the true number of clusters.
K-means can be effective for learning mixture models under certain conditions.
problem Learning the correct clustering of samples from a mixture of distributions.
method Optimizing the sum-of-squares distance between points and cluster centers, considering spherical Gaussian and log-concave distributions.
result Optimal clusterings are close to the correct target clustering under specific conditions.
SGMM clusters data in one pass, reducing dimensionality.
problem Clustering high-dimensional data efficiently.
method One-pass sparsification of Gaussian mixture model.
result SGMM achieves clustering accuracy with reduced computational cost.
Improved sample complexity for Gaussian Mixture Models using Pair Correlation Factor.
problem Understanding the sample complexity of Gaussian Mixture Models.
method Introducing Pair Correlation Factor (PCF) to measure clustering of component means and improving sample complexity bounds.
result The Pair Correlation Factor (PCF) more accurately determines the difficulty of parameter recovery in Gaussian Mixture Models.
A new method estimates the number of clusters on spherical data.
problem Estimating the number of clusters in spherical data.
method Spherical X-means (SX-means) method assuming von Mises-Fisher distributions.
result Shows the performance of SX-means in estimating the number of clusters.
Although many convex relaxations of clustering have been proposed in the past decade, current formulations remain restricted to spherical Gaussian or discriminative models and are susceptible to imbalanced clusters. To address these shortcomings, we propose a new class of convex relaxations that can be flexibly applied…
This paper studies clustering and embedding in high-dimensional Gaussian mixture block models.
problem Clustering and embedding in high-dimensional Gaussian mixture block models.
method Spectral clustering and embedding algorithms for graphs sampled from Gaussian mixture block models.
result Performance analysis of spectral clustering and embedding algorithms for 2-component spherical Gaussian mixtures.
The paper corrects for node degree in spectral clustering using random walk Laplacian.
problem Node degree heterogeneity in spectral clustering.
method Graph spectral embedding using the random walk Laplacian.
result The embedding provides uniformly consistent estimates of degree-corrected latent positions.
New clustering method for exponential family data.
problem Improving clustering for non-Gaussian data.
method Bregman Power k-Means algorithm.
result Outperforms existing methods in non-Gaussian data settings.
Paper learns mixture of Gaussians from streaming data.
problem Learning a mixture of Gaussians from a stream of data.
method Streaming version of Lloyd's heuristic, PCA-based seeding.
result Estimates centers of Gaussians accurately if sufficiently separated.
Proposes spherical text embedding for better directional similarity.
problem Directional similarity is more effective but unsupervised text embeddings are typically learned in Euclidean space.
method Develops a spherical generative model and an efficient optimization algorithm for unsupervised word and paragraph embeddings.
result Achieves state-of-the-art performances on various text embedding tasks.
Proposes a method to decompose multivariate signals into Gaussian components.
problem Decomposing multivariate signals into Gaussian components.
method Greedy variational method for non-negative multivariate signals as a weighted sum of Gaussians.
result Upper bound for the distance from any mode of a Gaussian mixture model to the set of corresponding means.
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. Robustly clusters mixtures of Gaussians even with outliers.
problem Clustering mixtures of statistically separated Gaussians robustly to outliers.
method Uses certifiable hypercontractivity, bounded variance, and anti-concentration of linear projections.
result First efficient algorithm for robust clustering of statistically separated Gaussians mixtures.
The study bounds the stability of Gaussian mixtures under small perturbations.
problem Stability of Gaussian mixtures under small changes in distribution.
method Deriving an explicit bound on parameter stability of spherical Gaussian Mixture Models (sGMM) in a pre-defined model class.
result Upper bound on parameter distance of close sGMMs to the original sGMM, dependent only on the original model.
Proposes variational Wasserstein barycenters for geometric clustering.
problem Geometric clustering problems, especially K-means and co-clustering.
method Solves for Monge maps using variational principle, explores connections to K-means and co-clustering.
result Demonstrates feasibility and use of variational Wasserstein barycenters in clustering.
Revisits Gaussian process model with spherical harmonics for scalable deep learning.
problem Scaling Gaussian process models to large input dimensions with high frequency learning.
method Introduces new kernels related to deep models, variational learning of spherical harmonic phases, and sparseness in eigenbasis.
result Enables scaling to larger input dimensions and learning of high frequency variations.
Software package assesses spherical data distributions and clusters.
problem Assessing and clustering spherical data distributions.
method Innovative goodness-of-fit tests and clustering algorithms using kernel-based quadratic distances.
result Efficient and mathematically sound goodness-of-fit tests for spherical data.
New GMM models fit high-dimensional data with fewer parameters.
problem Overparameterization and lack of flexibility in GMMs for high-dimensional data.
method Piecewise-constant covariance eigenvalue profiles, EM and penalized EM algorithms.
result Superior likelihood-parsimony tradeoffs in density fitting, clustering, and denoising.
New spectral clustering method for graphs with uneven node degrees.
problem Challenges in community detection for graphs with heterogeneous degree distributions.
method Spectral clustering on spherical coordinates with degree correction.
result Improved performance in representing computer networks.
Corrected whitening restores orthogonality in high-dimensional spherical Gaussian mixtures.
problem In high-dimensional data, standard whitening fails to preserve orthogonality of mixture means.
method Derived exact limits for whitened means dot products using random matrix theory, constructed a corrected whitening matrix.
result Corrected whitening allows for improved estimation of spherical Gaussian mixtures in the large-dimensional regime.
Existence and uniqueness of spherical helicoidal surfaces in 3-sphere via spherical curves.
problem Existence and uniqueness of spherical helicoidal surfaces in 3-sphere.
method Continuous function of distance to axis, spherical angular momentum of spherical curves.
result Existence and uniqueness theorem for spherical helicoidal surfaces in 3-sphere.
Bayesian approach approximates probability functions of Gaussian mixtures.
problem Approximating probability functions of non-spherical Gaussian mixtures.
method Bayesian decomposition, spherical radial decomposition, random sampling.
result Established differentiability and integral representation of gradient for probability functions.
Efficient algorithm learns mixture models of heavy-tailed distributions.
problem Learning mixture models of heavy-tailed distributions.
method Efficient high-dimensional sparse Fourier transforms.
result Algorithm succeeds for heavy-tailed distributions, including Laplace but excluding Gaussians.
Lloyd's K-means is shown to be a Frank-Wolfe algorithm variant.
problem Optimizing the sum of squared errors in clustering.
method Established a connection between Lloyd's K-means and Frank-Wolfe algorithm, derived convergence rates, and developed FW variants for empty clusters.
result Lloyd's K-means is a special case of the Frank-Wolfe algorithm with a non-asymptotic convergence rate of O(1/t).
Hybrid clustering merges K-means and hierarchical methods for diverse group shapes.
problem Clustering homogeneous spherical groups in large datasets.
method First, K-means partitions the dataset into spherical groups. Then, hierarchical clustering merges these groups with a data-driven distance measure. result Hybrid approach reveals general-shaped groups in datasets.
Researchers found multiple spherical Ricci metrics on tori with rotational symmetry.
problem Constructing and analyzing spherical Ricci metrics with rotational symmetry.
method Explicitly constructed a two-parameter family of metrics with rotational symmetry and showed their existence on tori.
result Infinitely many non-isometric spherical Ricci metrics can be realized on the same torus.
A new GP model uses spherical harmonics for faster inference.
problem Efficiently fitting large datasets with Gaussian processes.
method Sparse Gaussian processes with spherical harmonic features.
result Significant speed-up in inference for large datasets.
Proves a special case of the Gaussian kinematic formula using large sphere limits.
problem Proving a special case of the Gaussian kinematic formula.
method Viewing the GKF as the limit of spherical kinematic formulas for large dimension spheres.
result Proves a special case of the Gaussian kinematic formula.
Enhances Gaussian processes with spherical features for better scalability and flexibility.
problem Lack of representation learning in Gaussian processes compared to deep neural networks.
method Introduces spherical inter-domain features to improve GP approximation and scalability.
result The method alleviates limitations and improves scalability compared to alternative strategies.
Bayesian framework for sphere regression using Gaussian fields.
problem Nonparametric regression on the sphere with Gaussian priors.
method Isotropic Gaussian field priors, harmonic structure, exact posterior distributions, optimal spectral truncation, posterior contraction rates.
result Sharp posterior contraction rates for Gaussian priors with polynomially decaying angular power spectra.
A novel method relaxes binary constraints to non-negative spheres for multi-matching and clustering.
problem Optimization problems over binary matrices with injectivity constraints.
method Non-negative spherical relaxation followed by conditional power iteration.
result Automatic adjustment of the continuous parameter related to universe size.
Develops efficient algorithms for learning latent-variable models using implicit moment tensor computation.
problem Learning latent-variable models with moment tensors of super-constant degree.
method Implicit moment tensor computation for general models, extending previous work on clustering mixtures of spherical Gaussians.
result First poly(d, k) time learning algorithms for various models including mixtures of linear regressions, spherical Gaussians, and positive linear combinations of non-linear activations.
This work provides a computationally efficient and statistically consistent moment-based estimator for mixtures of spherical Gaussians. Under the condition that component means are in general position, a simple spectral decomposition technique yields consistent parameter estimates from low-order observable moments, wit…
Modified Epanechnikov Mean Shift converges to cluster centroids.
problem Lack of theoretical support for convergence of Epanechnikov Mean Shift due to non-smooth kernel density functions.
method Proposed a simple remedy to fix convergence issues, ensuring termination at a local maximum of the estimated density.
result Modified Epanechnikov Mean Shift guarantees convergence to a cluster centroid within a finite number of iterations.
Exact cluster recovery with same-cluster queries for arbitrary ellipsoidal clusters.
problem Recovering clusters from same-cluster queries in arbitrary ellipsoidal clusters.
method Relaxing spherical k-means assumption to arbitrary ellipsoidal clusters, designing an algorithm with logarithmic query complexity. result Exact recovery of clusters using O(k3lnklnn) queries and ildeO(kn+k3) time. Algorithm distinguishes Gaussian mixtures from pure Gaussians in quasi-polynomial time.
problem Distinguishing mixtures of Gaussian components from pure Gaussians, especially when components are well-separated.
method Sum-of-Squares method, quasi-polynomial time algorithm, bipartitioning sample to separate components.
result Algorithm can reliably distinguish between mixtures and pure Gaussians in quasi-polynomial time.
Study on generalized spherical surfaces in Euclidean spaces.
problem Characterizing and analyzing generalized spherical surfaces in Euclidean spaces.
method Investigation of generalized spherical curves and surfaces in Euclidean spaces, calculation of curvatures.
result Identification and characterization of different types of generalized spherical surfaces in Euclidean spaces.
Study singularities of constant curvature surfaces and harmonic maps.
problem Understanding singularities and bifurcations of constant curvature surfaces.
method Loop group methods to construct bifurcations and analyze harmonic maps.
result Determine which map germs can be represented by harmonic maps.
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.
Study finds the cutoff for exact recovery in Gaussian mixture models.
problem Determining the separation of cluster centers for exact recovery in Gaussian mixture models.
method Used information theory and SDP relaxation of K-means clustering. result Sharp threshold for exact recovery of cluster labels without assuming cluster center symmetry.
Study spherical cap packing with probabilistic methods for detecting low-rank structures.
problem Detecting low-rank structures in high-dimensional Gaussian data.
method Probabilistic spherical cap packing approach for asymptotic bounds and extreme value distributions.
result Developed fast detection method for low-rank structures without spectrum information.
Minimal Lagrangian diffeomorphisms between spherical surfaces are rigid.
problem Rigidity of minimal Lagrangian diffeomorphisms between spherical surfaces.
method Proving that any minimal Lagrangian diffeomorphism between two closed spherical surfaces with cone singularities is an isometry.
result Minimal Lagrangian diffeomorphisms between spherical surfaces are rigid (i.e., they are isometries).
Study free energy in spherical spin glasses, proving universality dichotomy.
problem Analyzing free energy in spherical spin glass models with different tail exponents.
method Introduced a tail-adapted normalization and used universality dichotomy.
result Sharp universality dichotomy for free energy across different tail exponents.
We treat the problem of estimation of orientation parameters whose values are invariant to transformations from a spherical symmetry group. Previous work has shown that any such group-invariant distribution must satisfy a restricted finite mixture representation, which allows the orientation parameter to be estimated u…
A new vine copula mixture model improves clustering accuracy for non-Gaussian data.
problem Finite mixture models struggle with asymmetric tail dependencies and non-elliptical clusters.
method Proposes a vine copula mixture model for clustering non-Gaussian data, addressing model selection and parameter estimation.
result Significant improvement in clustering accuracy for data with asymmetric tail dependencies or non-Gaussian margins.