Related papers: Clustering in typical unit-distance avoiding sets
This paper studies the subspace clustering problem in which data points collected from high-dimensional ambient space lie in a union of linear subspaces. Subspace clustering becomes challenging when the dimension of intersection between…
In general, the clustering problem is NP-hard, and global optimality cannot be established for non-trivial instances. For high-dimensional data, distance-based methods for clustering or classification face an additional difficulty, the…
We introduce and study the $k$-center clustering problem with set outliers, a natural and practical generalization of the classical $k$-center clustering with outliers. Instead of removing individual data points, our model allows discarding…
We study the classic $k$-means/median clustering, which are fundamental problems in unsupervised learning, in the setting where data are partitioned across multiple sites, and where we are allowed to discard a small portion of the data by…
It is well known that almost every dilation of a sequence of real numbers, that diverges to $\infty$, is dense modulo~1. This paper studies the exceptional set of points -- those for which the dilation is not dense. Specifically, we…
We address the problem of un-supervised soft-clustering called micro-clustering. The aim of the problem is to enumerate all groups composed of records strongly related to each other, while standard clustering methods separate records at…
We improve the best known upper bound on the number of edges in a unit-distance graph on $n$ vertices for each $n\in\{16,\ldots,30\}$. When $n\leq 21$, our bounds match the best known lower bounds, and we fully enumerate the densest…
We present a new approach to the problem of enumerating permutations of length n that avoid a fixed consecutive pattern of length m. We use this idea to give explicit upper and lower bounds on the number of permutations avoiding a pattern…
A question going back to Halmos asks when two approximately commuting matrices of a certain kind are close to exactly commuting matrices of the same kind. It has long been known that there is a winding number obstruction for approximately…
We prove a generalization of the author's work to show that any subset of the primes which is `well-distributed' in arithmetic progressions contains many primes which are close together. Moreover, our bounds hold with some uniformity in the…
Clustering is a widely used technique with a long and rich history in a variety of areas. However, most existing algorithms do not scale well to large datasets, or are missing theoretical guarantees of convergence. This paper introduces a…
By virtue of their high galaxy space densities and their large spatial separations, clusters are efficient and accurate tracers of the large-scale density and velocity fields. Substantial progress has been made over the past decade in the…
The observed incidence of strongly lensing clusters exceeds the predictions of a Lambda-CDM model by about a factor of 10. We revisit the observational side of this discrepancy by measuring the incidence of strong lensing in a subsample of…
The problem of finding groups in data (cluster analysis) has been extensively studied by researchers from the fields of Statistics and Computer Science, among others. However, despite its popularity it is widely recognized that the…
Spectral clustering is one of the most prominent clustering approaches. The distance-based similarity is the most widely used method for spectral clustering. However, people have already noticed that this is not suitable for multi-scale…
In the context of clustering, we assume a generative model where each cluster is the result of sampling points in the neighborhood of an embedded smooth surface; the sample may be contaminated with outliers, which are modeled as points…
Density-based clustering relies on the idea of linking groups to some specific features of the probability distribution underlying the data. The reference to a true, yet unknown, population structure allows to frame the clustering problem…
We present a new clustering method in the form of a single clustering equation that is able to directly discover groupings in the data. The main proposition is that the first neighbor of each sample is all one needs to discover large chains…
The singular subspaces perturbation theory is of fundamental importance in probability and statistics. It has various applications across different fields. We consider two arbitrary matrices where one is a leave-one-column-out submatrix of…
In this paper, we address an issue of finding explainable clusters of class-uniform data in labelled datasets. The issue falls into the domain of interpretable supervised clustering. Unlike traditional clustering, supervised clustering aims…