Related papers: Intrinsic Dimension of Geometric Data Sets
Manifold hypothesis states that data points in high-dimensional space actually lie in close vicinity of a manifold of much lower dimension. In many cases this hypothesis was empirically verified and used to enhance unsupervised and…
Knowledge graph embeddings (KGE) have been extensively studied to embed large-scale relational data for many real-world applications. Existing methods have long ignored the fact many KGs contain two fundamentally different views: high-level…
We study the Hausdorff measure and dimension of the set of intrinsically simultaneously $\psi$-approximable points on a curve, surface, etc., given as a graph of integer valued polynomials. We obtain complete answers to these questions for…
The concentration of measure phenomena were discovered as the mathematical background of statistical mechanics at the end of the XIX - beginning of the XX century and were then explored in mathematics of the XX-XXI centuries. At the…
In various areas of modern physics and in particular in quantum gravity or foundational space-time physics it is of great importance to be in the possession of a systematic procedure by which a macroscopic or continuum limit can be…
In the manifold learning problem one seeks to discover a smooth low dimensional surface, i.e., a manifold embedded in a higher dimensional linear vector space, based on a set of measured sample points on the surface. In this paper we…
Deep metric learning (DML) is a cornerstone of many computer vision applications. It aims at learning a mapping from the input domain to an embedding space, where semantically similar objects are located nearby and dissimilar objects far…
Entropy is useful in statistical problems as a measure of irreversibility, randomness, mixing, dispersion, and number of microstates. However, there remains ambiguity over the precise mathematical formulation of entropy, generalized beyond…
Geometric ability is a significant challenge for large language models (LLMs) due to the need for advanced spatial comprehension and abstract thinking. Existing datasets primarily evaluate LLMs on their final answers, but they cannot truly…
For any finite point set in $D$-dimensional space equipped with the 1-norm, we present random linear embeddings to $k$-dimensional space, with a new metric, having the following properties. For any pair of points from the point set that are…
This report concerns the problem of dimensionality reduction through information geometric methods on statistical manifolds. While there has been considerable work recently presented regarding dimensionality reduction for the purposes of…
Exploiting internal spatial geometric constraints of sparse LiDARs is beneficial to depth completion, however, has been not explored well. This paper proposes an efficient method to learn geometry-aware embedding, which encodes the local…
The asymptotic dimension is an invariant of metric spaces introduced by Gromov in the context of geometric group theory. In this paper, we study the asymptotic dimension of metric spaces generated by graphs and their shortest path metric…
The manifold hypothesis suggests that high-dimensional data often lie on or near a low-dimensional manifold. Estimating the dimension of this manifold is essential for leveraging its structure, yet existing work on dimension estimation is…
The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer…
Manifold learning has been proven to be an effective method for capturing the implicitly intrinsic structure of non-Euclidean data, in which one of the primary challenges is how to maintain the distortion-free (isometry) of the data…
The advent of modern technology, permitting the measurement of thousands of characteristics simultaneously, has given rise to floods of data characterized by many large or even huge datasets. This new paradigm presents extraordinary…
Dimensionality reduction is a fundamental task that aims to simplify complex data by reducing its feature dimensionality while preserving essential patterns, with core applications in data analysis and visualisation. To preserve the…
Metric magnitude is a measure of the "size" of point clouds with many desirable geometric properties. It has been adapted to various mathematical contexts and recent work suggests that it can enhance machine learning and optimization…
In this paper geometry is studied with a novel approach. Every geometrical object is defined as a symbol which satisfies some properties. These symbols are then coded into a class of numbers which are named here as many dots numbers (MDN).…