Related papers: Clustering in Hilbert simplex geometry

In this work, we introduce a novel methodology for divisive hierarchical clustering. Our divisive (``top-down'') approach is motivated by the fact that agglomerative hierarchical clustering (``bottom-up''), which is commonly used for…

Clustering is indispensable for data analysis in many scientific disciplines. Detecting clusters from heavy noise remains challenging, particularly for high-dimensional sparse data. Based on graph-theoretic framework, the present paper…

We consider a system with a discrete configuration space. We show that the geometrical structures associated with such a system provide the tools necessary for a reconstruction of discrete quantum mechanics once dynamics is brought into the…

Modern datasets across many disciplines increasingly consist of time-evolving, potentially infinite-dimensional random objects, such as dynamic functional data, which are naturally modeled in Hilbert spaces. In these settings,…

In machine learning, observation features are measured in a metric space to obtain their distance function for optimization. Given similar features that are statistically sufficient as a population, a statistical distance between two…

The space of probability distributions on a given sample space possesses natural geometric properties. For example, in the case of a smooth parametric family of probability distributions on the real line, the parameter space has a…

Embedding complex objects as vectors in low dimensional spaces is a longstanding problem in machine learning. We propose in this work an extension of that approach, which consists in embedding objects as elliptical probability…

We develop a novel clustering method for distributional data, where each data point is regarded as a probability distribution on the real line. For distributional data, it has been challenging to develop a clustering method that utilizes…

In this work clustering schemes for uncertain and structured data are considered relying on the notion of Wasserstein barycenters, accompanied by appropriate clustering indices based on the intrinsic geometry of the Wasserstein space where…

Although distance measures are used in many machine learning algorithms, the literature on the context-independent selection and evaluation of distance measures is limited in the sense that prior knowledge is used. In cluster analysis,…

This paper concerns models and convergence principles for dealing with stochasticity in a wide range of algorithms arising in nonlinear analysis and optimization in Hilbert spaces. It proposes a flexible geometric framework within which…

Collections of probability distributions arise in a variety of applications ranging from user activity pattern analysis to brain connectomics. In practice these distributions can be defined over diverse domain types including finite…

This paper proposes a hierarchical clustering approach for the segmentation of mobile LiDAR point clouds. We perform the hierarchical clustering on unorganized point clouds based on a proximity matrix. The dissimilarity measure in the…

This paper considers metric spaces where distances between a pair of nodes are represented by distance intervals. The goal is to study methods for the determination of hierarchical clusters, i.e., a family of nested partitions indexed by a…

Clustering is a fundamental analysis tool aiming at classifying data points into groups based on their similarity or distance. It has found successful applications in all natural and social sciences, including biology, physics, economics,…

Two geometrical structures have been extensively studied for a manifold of probability distributions. One is based on the Fisher information metric, which is invariant under reversible transformations of random variables, while the other is…

Clustering is one of the most fundamental tasks in machine learning, and the k-means clustering algorithm is perhaps one of the most widely used clustering algorithms. However, it suffers from several limitations, such as sensitivity to…

We show that the Kullback-Leibler distance is a good measure of the statistical uncertainty of correlation matrices estimated by using a finite set of data. For correlation matrices of multivariate Gaussian variables we analytically…

Most research into similarity search in metric spaces relies upon the triangle inequality property. This property allows the space to be arranged according to relative distances to avoid searching some subspaces. We show that many common…

This paper deals with clustering methods based on adaptive distances for histogram data using a dynamic clustering algorithm. Histogram data describes individuals in terms of empirical distributions. These kind of data can be considered as…