Related papers: Geometry of Similarity Comparisons
As a rule statistical measures are often vulnerable to the presence of outliers and spatial correlation coefficients, critical in the assessment of spatial data, remain susceptible to this inherent flaw. In contexts where data originates…
In this work, we develop new generalization bounds for neural networks trained on data supported on Riemannian manifolds. Existing generalization theories often rely on complexity measures derived from Euclidean geometry, which fail to…
The metric dimension of a graph is the smallest number of nodes required to identify all other nodes based on shortest path distances uniquely. Applications of metric dimension include discovering the source of a spread in a network,…
For pattern recognition like image recognition, it has become clear that each machine-learning dictionary data actually became data in probability space belonging to Euclidean space. However, the distances in the Euclidean space and the…
Measuring the similarity between data points often requires domain knowledge, which can in parts be compensated by relying on unsupervised methods such as latent-variable models, where similarity/distance is estimated in a more compact…
For two closed curves on a plane (discrete version) and local criteria for similarity of points on the curves one gets a potential, which describes the similarity between curve points. This is the base for a global similarity measure of…
The optimal Orlicz target space is exhibited for embeddings of fractional-order Orlicz-Sobolev spaces in $\mathbb R^n$. An improved embedding with an Orlicz-Lorentz target space, which is optimal in the broader class of all…
There are many methods developed to approximate a cloud of vectors embedded in high-dimensional space by simpler objects: starting from principal points and linear manifolds to self-organizing maps, neural gas, elastic maps, various types…
Dimension is a fundamental property of objects and the space in which they are embedded. Yet ideal notions of dimension, as in Euclidean spaces, do not always translate to physical spaces, which can be constrained by boundaries and…
Analogue gravity is based on a mathematical identity between quantum field theory in curved space-time and the propagation of perturbations in certain condensed matter systems. But not every curved space-time can be simulated in such a way,…
In the era of foundation models and Large Language Models (LLMs), Euclidean space has been the de facto geometric setting for machine learning architectures. However, recent literature has demonstrated that this choice comes with…
The methods of determining the fractal dimension and irregularity scale in simulated galaxy catalogs and the application of these methods to the data of the 2dF and 6dF catalogs are analyzed. Correlation methods are shown to be correctly…
Distance measuring is a very important task in digital geometry and digital image processing. Due to our natural approach to geometry we think of the set of points that are equally far from a given point as a Euclidean circle. Using the…
For a large class of self-similar sets F in R^d analogues of the higher order mean curvatures of differentiable submanifolds are introduced, in particular, the fractal Gauss-type curvature. They are shown to be the densities of associated…
The statistical analysis of data lying on a differentiable, locally Euclidean, manifold introduces a variety of challenges because the analogous measures to standard Euclidean statistics are local, that is only defined within a…
Metric search is concerned with the efficient evaluation of queries in metric spaces. In general,a large space of objects is arranged in such a way that, when a further object is presented as a query, those objects most similar to the query…
In the elastic shape analysis approach to shape matching and object classification, plane curves are represented as points in an infinite-dimensional Riemannian manifold, wherein shape dissimilarity is measured by geodesic distance. A…
Diameter, radius and eccentricities are fundamental graph parameters, which are extensively studied in various computational settings. Typically, computing approximate answers can be much more efficient compared with computing exact…
A new class of distances appropriate for measuring similarity relations between sequences, say one type of similarity per distance, is studied. We propose a new ``normalized information distance'', based on the noncomputable notion of…
We extend the Theory of Computation on real numbers, continuous real functions, and bounded closed Euclidean subsets, to compact metric spaces $(X,d)$: thereby generically including computational and optimization problems over higher types,…