The paper proves a theorem linking convex body centroids and category theory.
problem Understanding centroids of sections of convex bodies.
method Lusternik-Schnirelmann category theory.
result At least n hyperplanes exist such that the center of mass of their intersection with a convex body lies on the boundary of the convex body.
The paper generalizes the second Pappus-Guldin theorem for calculating volumes of bodies.
problem Calculating the volume of a body cut into perpendicular slices.
method Using a generalized formula and properties of centroids and floating bodies.
result A curve with centroid property exists for convex bodies, leading to simpler volume calculations.
Sharp Lp affine isoperimetric inequalities are established for the entire class of Lp projection bodies and the entire class of Lp centroid bodies. These new inequalities strengthen the Lp Petty projection and the Lp Busemann--Petty centroid inequality.
The paper explores centroids and static equilibrium points in non-Euclidean geometries.
problem Investigating centroids and static equilibrium points in spherical, hyperbolic, and normed spaces.
method Extending Gal'perin's work, the paper examines convex bodies in these spaces and analyzes the minimum number of equilibrium points.
result Every plane convex body in any of these spaces has at least four equilibrium points, and there are mono-monostatic convex bodies in 3D spherical, hyperbolic, and certain normed spaces.
We introduce a new volume definition on normed vector spaces. We show that the induced k-area functionals are convex for all k. In the particular case k=2, our theorem implies that Busemann's 2-volume density is convex, which was recently shown by Burago-Ivanov. We also show how the new volume definition is relat…
Optimal inequality on sphere for convex bodies.
problem Bounding the spherical measure of a convex body's intersection with a plane orthogonal to its centroid.
method Proving an inequality on the sphere using convex geometry.
result The inequality is optimal with a constant of \((1-1/n)^{n-1}\).
The study characterizes quadrics among affine hyperspheres based on centroid collinearity of sections.
problem Characterizing quadrics among affine hyperspheres based on section centroid collinearity.
method Extending Meyer and Reisner's theorem to unbounded convex sets and identifying additional assumptions.
result Ellipsoids, paraboloids, and one sheet of a two-sheeted hyperboloid are the only quadrics satisfying the centroid collinearity condition.
New convex body defined to prove Busemann Random Simplex Inequality.
problem Proving Busemann Random Simplex Inequality for general p.
method Defining NpL and proving isoperimetric inequality for its dual. result Equivalent to Lp-Busemann-Petty centroid inequality for p=1,2. We show that the cone-volume measure of a convex body with centroid at the origin satisfies the subspace concentration condition. This implies, among others, a conjectured best possible inequality for the U-functional of a convex body. For both results we provide stronger versions in the sense of stability i…
Study on Santaló point for convex bodies in normed spaces.
problem Exploring Santaló point for convex bodies in normed spaces.
method Existence and uniqueness proof for C1 norms, dual Santaló point for smooth curved unit balls. result Existence and uniqueness of Santaló point for convex bodies in normed spaces.
In [Centro-affine invariants for smooth convex bodies, Int. Math. Res. Notices. doi: 10.1093/imrn/rnr110, 2011] Stancu introduced a family of centro-affine normal flows, p-flow, for 1≤p<∞. Here we investigate the asymptotic behavior of the planar p-flow for p=∞ in the class of smooth, origin-symme…
New algorithm HTICA solves heavy-tailed ICA problems efficiently.
problem Heavy-tailed data in ICA problems.
method Uses centroid body and explicit analytic representations to bypass ellipsoid method and random walks.
result Outperforms other algorithms in heavy-tailed regimes on real and synthetic data.
The paper extends inequalities for convex bodies to higher dimensions and various norms.
problem Extending inequalities for convex bodies to higher dimensions and various norms.
method Developed new operators and inequalities for higher-order Lp norms. result Established mth-order Lp isoperimetric inequalities. New method for k-modes algorithm improves clustering performance.
problem Improving initial solution selection for k-modes algorithm.
method Uses Hospital-Resident Assignment Problem to find initial cluster centroids.
result Outperforms other initialisations in most cases, especially for low-density data.
mmFall detects falls using mmWave radar and a hybrid VRAE, achieving high accuracy.
problem Detecting falls accurately and privately using mmWave radar.
method Uses mmWave radar for body point cloud and centroid, combined with a hybrid VRAE for anomaly detection.
result Achieves 98% fall detection with 2 false alarms out of 50 falls.
New proof shows origin-centred balls are unique solutions to curvature problems.
problem Proving uniqueness of solutions to curvature problems.
method Local Brunn-Minkowski inequality and Alexandrov-Fenchel inequality.
result Origin-centred balls are the only solutions to curvature and related problems.
A new method for person recognition using cosine loss.
problem Recognizing the same identity across time and space with complicated scenes and similar appearance.
method Proposes a congenerous cosine loss to train a network for robust and representative features.
result The proposed method achieves better classification accuracy than previous state-of-the-arts.
New method improves text classification using KL divergence.
problem Multinomial text classification problem
method Centroid estimation based on symmetric KL divergence
result Substantial improvements over traditional classifiers
CDF uses centroids to split features for high-dimensional classification.
problem High-dimensional classification problems with complex class structures.
method CDF introduces a centroid-driven splitting strategy in decision trees.
result CDF outperforms conventional methods in high-dimensional classification.
A new classification method using disjoint centroids and normalized distance.
problem Improving classification accuracy and feature selection.
method Nearest disjoint centroid classifier with normalized distance.
result Our method outperforms other classifiers in terms of misclassification rates and feature usage.
Sparse nearest-centroid classifiers detect relevant features for classification.
problem Classifying data with low computational cost and feature selection.
method Proposes ℓ1 and ℓ2 sparse variants of nearest-centroid classifiers. result Training sparse classifiers can be done exactly and at quasi-linear cost.
Ball k-means reduces point-centroid distance computations for faster k-means clustering.
problem Efficiently finding k-means clusters in large datasets.
method Uses a ball to describe clusters, dividing them into stable and active areas, and adjusting points within annulus areas.
result Significantly reduces point-centroid distance computations, making k-means faster and more efficient.
Improved image classification using centroids and stochastic sampling.
problem Limited accuracy of nearest-neighbor classification.
method Coarse-graining (replacing images by centroids) and stochastic sampling of centroids.
result Stochastic sampling of centroids improves classification accuracy.
Centroid Transformers reduce memory and computation by summarizing inputs into centroids.
problem Efficiently summarize inputs with reduced memory and computation.
method Generalizes self-attention to map N inputs to M centroids (M ≤ N), reducing complexity.
result Centroid Transformers reduce memory and computation while preserving key information.
EKM addresses imbalanced data clustering by repelling centroids in large clusters.
problem Imbalanced data leads to biased clustering of large clusters.
method EKM introduces a novel centroid repulsion mechanism based on the Boltzmann operator.
result EKM outperforms benchmark algorithms on imbalanced data.
Centroid-Encoder reduces high-dimensional data for better visualization.
problem Visualizing high-dimensional data efficiently and accurately.
method Centroid-Encoder integrates label information to keep similar objects close in reduced space.
result Centroid-Encoder outperforms other techniques in visualizing high-dimensional data.
Empty core found in max-loss non-centroid clustering.
problem Core stability in non-centroid clustering under max-loss objective.
method Proof for all k≥3 and n≥9 agents, computer-aided proof for 2D Euclidean points.
result Core can be empty in non-centroid clustering under max-loss objective.
Softmax and k-means clustering are mathematically linked, improving neural network robustness.
problem Improving neural network robustness against adversarial attacks.
method Formally proving the connection between softmax and k-means, proposing Centroid Based Tailoring.
result The proposed Gauss network is less susceptible to one-pixel attacks.
Due to the success of the bag-of-word modeling paradigm, clustering histograms has become an important ingredient of modern information processing. Clustering histograms can be performed using the celebrated k-means centroid-based algorithm. From the viewpoint of applications, it is usually required to deal with symm…
Centroids Matching tackles catastrophic forgetting by matching feature vectors to class centroids.
problem Catastrophic forgetting in neural networks when learning new tasks.
method Centroids Matching operates in the embedding space of neural network features, matching these vectors to class centroids.
result Centroids Matching achieves high accuracy on all tasks without using external memory, even in realistic scenarios.
Optimizes a small set of centroid points to approximate bootstrap distribution.
problem Computational inefficiency of standard bootstrap methods in large-scale machine learning.
method Explicitly optimizes a small set of high quality centroid points to approximate the ideal bootstrap distribution.
result Accurately estimates uncertainty with a small number of bootstrap centroids, outperforming i.i.d. sampling.
Text clustering method replaces centroids with summaries for interpretability and scalability.
problem Efficiently clustering text data while maintaining interpretability and scalability.
method k-NLPmeans and k-LLMmeans, which periodically replace numeric centroids with textual summaries.
result Consistently outperforms classical baselines and recent LLM-based clustering methods.
In addition to finding meaningful clusters, centroid-based clustering algorithms such as K-means or mean-shift should ideally find centroids that are valid patterns in the input space, representative of data in their cluster. This is challenging with data having a nonconvex or manifold structure, as with images or text…
Optimizes K-means clustering with PSO for better accuracy.
problem Improving the accuracy of K-means clustering.
method Uses Particle Swarm Optimization (PSO) to find optimal initial centroids for K-means.
result Optimal centroids found using PSO lead to better clustering accuracy.
CCC clusters with controlled spread, outperforming standard methods.
problem Compact and structured clustering with spread control.
method Constrained Centroid Clustering using Lagrangian formulation.
result CCC achieves more compact clusters with reduced radial spread.
New method speeds up CNN inference by storing centroids instead of full filters.
problem Reduction of time and space complexity in CNN inference.
method Centroid filter quantization for convolutional layers.
result 2.9 times better computational gain on ImageNet benchmark.
We study a notion of a Lipschitz, permutation-invariant "centroid" for triples of points in mapping class groups MCG(S), which satisfies a certain polynomial growth bound. A consequence (via work of Drutu-Sapir or Chatterji-Ruane) is the Rapid Decay Property for MCG(S).
Method counters noisy labels by discounting distant samples.
problem Training models with noisy labels in medical and autonomous domains.
method Discounting distant samples from class centroids in latent space.
result Significant improvements in classification accuracy.
New meta-learning method improves domain generalization by balancing parameters closer to domain centroids.
problem Improving domain generalization by reducing overfitting to specific domains.
method Arithmetic meta-learning with arithmetic-weighted gradients to balance parameters closer to domain centroids.
result Experimental validation of improved domain generalization performance.
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.
SIVF k-means algorithm speeds up sparse data clustering.
problem Efficiently clustering large-scale high-dimensional sparse data.
method Inverted-file structure for centroids, filter-based similarity reduction.
result SIVF achieves higher speed and lower memory consumption.
A semi-supervised learning method using predefined class centroids for image classification.
problem Reducing the need for labeled data in deep learning.
method Use a small number of labeled samples and data augmentation on unlabeled samples. Constrain all samples to predefined evenly-distributed class centroids (PEDCC) using loss functions.
result Achieves state-of-the-art results with minimal labeled data.
Proportional centroid clustering aims to fairly group points without prior protected subsets.
problem Fairly group points without prior protected subsets.
method Define fairness as proportionality, present algorithms for efficient computation and optimization.
result Proportional solutions trade off with the k-means objective.
This study evaluates cluster search algorithms using Gaussian mixture models.
problem Determining the optimal number of clusters in data sets generated by Gaussian mixture models.
method Examined centroid- and model-based cluster search algorithms in various cases.
result Model-based algorithms are more robust to cluster overlap and covariance type than centroid-based methods.
Novel method classifies HIV patients based on viral load patterns.
problem Limited methods classify patients by viral load patterns, often specific to study design.
method Four features, centroid-based classification algorithm, radial normalization classification.
result Classifies 1,576 HIV positive clinic patients into five viral load patterns.
Paper presents robust clustering methods for general mixture models.
problem Clustering with sub-Gaussian error assumptions often invalid in practice.
method Hybrid clustering with robust centroid estimate and data-driven initialization.
result Provably near-optimal mislabeling guarantees for general error distributions.
Multilayer bootstrap network builds a gradually narrowed multilayer nonlinear network from bottom up for unsupervised nonlinear dimensionality reduction. Each layer of the network is a nonparametric density estimator. It consists of a group of k-centroids clusterings. Each clustering randomly selects data points with r…
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.