Related papers: Properties of Algorithmic Information Distance
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…
Distances are pervasive in machine learning. They serve as similarity measures, loss functions, and learning targets; it is said that a good distance measure solves a task. When defining distances, the triangle inequality has proven to be a…
The unit Euclidean distance degree and the generic Euclidean distance degree are two well-studied invariants of projective varieties. These quantities measure the algebraic complexity of nearest-point problems on a variety, and in many…
We present a framework for embedding graph structured data into a vector space, taking into account node features and topology of a graph into the optimal transport (OT) problem. Then we propose a novel distance between two graphs, named…
The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the image of a non-closed geodesic has 0 distance from the set of conical points.…
The widely applied k-means algorithm produces clusterings that violate our expectations with respect to high/low similarity/density and is in conflict with Kleinberg's axiomatic system for distance based clustering algorithms that…
It is well-understood that different algorithms, training processes, and corpora produce different word embeddings. However, less is known about the relation between different embedding spaces, i.e. how far different sets of embeddings…
We develop a new class of distances for objects including lines, hyperplanes, and trajectories, based on the distance to a set of landmarks. These distances easily and interpretably map objects to a Euclidean space, are simple to compute,…
We define a class of Euclidean distances on weighted graphs, enabling to perform thermodynamic soft graph clustering. The class can be constructed form the "raw coordinates" encountered in spectral clustering, and can be extended by means…
The class of Schoenberg transformations, embedding Euclidean distances into higher dimensional Euclidean spaces, is presented, and derived from theorems on positive definite and conditionally negative definite matrices. Original results on…
Effectively applying the K-means algorithm to clustering tasks with incomplete features remains an important research area due to its impact on real-world applications. Recent work has shown that unifying K-means clustering and imputation…
Combining the techniques of approximation algorithms and parameterized complexity has long been considered a promising research area, but relatively few results are currently known. In this paper we study the parameterized approximability…
Euclidean distance matrices (EDMs) are a major tool for localization from distances, with applications ranging from protein structure determination to global positioning and manifold learning. They are, however, static objects which serve…
Some graphs admit drawings in the Euclidean k-space in such a (natu- ral) way, that edges are represented as line segments of unit length. Such drawings will be called k dimensional unit distance representations. When two non-adjacent…
Given two sets of points $A$ and $B$ in a normed plane, we prove that there are two linearly separable sets $A'$ and $B'$ such that $\mathrm{diam}(A')\leq \mathrm{diam}(A)$, $\mathrm{diam}(B')\leq \mathrm{diam}(B)$, and $A'\cup B'=A\cup B.$…
We consider two disjoint sets of points. If at least one of the sets can be embedded into an Euclidean space, then we provide sufficient conditions for the two sets to be jointly embedded in one Euclidean space. In this joint Euclidean…
This paper explores hierarchical clustering in the case where pairs of points have dissimilarity scores (e.g. distances) as a part of the input. The recently introduced objective for points with dissimilarity scores results in every tree…
Sliced mutual information (SMI) is defined as an average of mutual information (MI) terms between one-dimensional random projections of the random variables. It serves as a surrogate measure of dependence to classic MI that preserves many…
Multidimensional scaling (MDS) is the act of embedding proximity information about a set of $n$ objects in $d$-dimensional Euclidean space. As originally conceived by the psychometric community, MDS was concerned with embedding a fixed set…
This book introduces the new research area of Geometric Data Science, where data can represent any real objects through geometric measurements. The first part of the book focuses on finite point sets. The most important result is a complete…