Related papers: The Remarkable Simplicity of Very High Dimensional…

In this paper we show that in systems where the probability distribution of the the overlap is non trivial in the infinity volume limit, the property of ultrametricity can be proved in general starting from two very simple and natural…

In recent years, topological data analysis has been utilized for a wide range of problems to deal with high dimensional noisy data. While text representations are often high dimensional and noisy, there are only a few work on the…

Network embedding has recently attracted lots of attentions in data mining. Existing network embedding methods mainly focus on networks with pairwise relationships. In real world, however, the relationships among data points could go beyond…

A classic problem in unsupervised learning and data analysis is to find simpler and easy-to-visualize representations of the data that preserve its essential properties. A widely-used method to preserve the underlying hierarchical structure…

Modern inference and learning often hinge on identifying low-dimensional structures that approximate large scale data. Subspace clustering achieves this through a union of linear subspaces. However, in contemporary applications data is…

Hyperbolic models are remarkably good at reproducing the scale-free, highly clustered and small-world properties of networks representing real complex systems in a very simple framework. Here we show that for the popularity-similarity…

This paper is about metric data structures in high-dimensional or non-Euclidean space that permit cached sufficient statistics accelerations of learning algorithms. It has recently been shown that for less than about 10 dimensions,…

We outline a novel clustering scheme for simplicial complexes that produces clusters of simplices in a way that is sensitive to the homology of the complex. The method is inspired by, and can be seen as a higher-dimensional version of,…

Subspace clustering is an unsupervised clustering technique designed to cluster data that is supported on a union of linear subspaces, with each subspace defining a cluster with dimension lower than the ambient space. Many existing…

Clustering $\unicode{x2013}$ the tendency for neighbors of nodes to be connected $\unicode{x2013}$ quantifies the coupling of a complex network to its latent metric space. In random geometric graphs, clustering undergoes a continuous phase…

Understanding the topological characteristics of data is important to many areas of research. Recent work has demonstrated that synthetic 4D image-type data can be useful to train 4D convolutional neural network models to see topological…

We introduce a supervised dimensionality reduction methodology for categorical (and discretized mixed-type) data based on a density-matrix construction induced by class-conditional frequencies. Given a labeled dataset encoded in a one-hot…

The rapid emergence of high-dimensional data in various areas has brought new challenges to current ensemble clustering research. To deal with the curse of dimensionality, recently considerable efforts in ensemble clustering have been made…

How to extract useful insights from data is always a challenge, especially if the data is multidimensional. Often, the data can be organized according to certain hierarchical structure that are stemmed either from data collection process or…

Hierarchical clustering is an effective, interpretable method for analyzing structure in data. It reveals insights at multiple scales without requiring a predefined number of clusters and captures nested patterns and subtle relationships,…

Ultametrics are an important class of distances used in applications such as phylogenetics, clustering and classification theory. Ultrametrics are essentially distances that can be represented by an edge-weighted rooted tree so that all of…

Homology-based invariants can be used to characterize the geometry of datasets and thereby gain some understanding of the processes generating those datasets. In this work we investigate how the geometry of a dataset changes when it is…

We propose a flexible and multi-scale method for organizing, visualizing, and understanding datasets sampled from or near stratified spaces. The first part of the algorithm produces a cover tree using adaptive thresholds based on a…

Consider a high-dimensional data set, in which for every data-point there is incomplete information. Each object in the data set represents a real entity, which is described by a point in high-dimensional space. We model the lack of…

Dimensionality is one of the most important properties of complex physical systems. However, only recently this concept has been considered in the context of complex networks. In this paper we further develop the previously introduced…