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A finite set of unlabelled points in Euclidean space is the simplest representation of many real objects from mineral rocks to sculptures. Since most solid objects are rigid, their natural equivalence is rigid motion or isometry maintaining…
In this manuscript, we show how conformal invariance can be incorporated in a classical theory of gravitation, in the context of metric measure space. Metric measure space involves a geometrical scalar $f$, dubbed as density function, which…
Distance metric learning is a branch of machine learning that aims to learn distances from the data, which enhances the performance of similarity-based algorithms. This tutorial provides a theoretical background and foundations on this…
Analysis of spatial multivariate data, i.e., measurements at irregularly-spaced locations, is a challenging topic in visualization and statistics alike. Such data are integral to many domains, e.g., indicators of valuable minerals are…
We consider the numbers of positive and negative eigenvalues of matrices of squared distances between randomly sampled i.i.d. points in a given metric measure space. These numbers and their limits, as the number of points grows, in fact…
We propose an iterative scheme for feature-based positioning using a new weighted dissimilarity measure with the goal of reducing the impact of large errors among the measured or modeled features. The weights are computed from the…
In recent years WiFi became the primary source of information to locate a person or device indoor. Collecting RSSI values as reference measurements with known positions, known as WiFi fingerprinting, is commonly used in various positioning…
Gromov-Wasserstein distance has found many applications in machine learning due to its ability to compare measures across metric spaces and its invariance to isometric transformations. However, in certain applications, this invariance…
The similarity between trajectory patterns in clustering has played an important role in discovering movement behaviour of different groups of mobile objects. Several approaches have been proposed to measure the similarity between sequences…
Understanding geometric properties of natural language processing models' latent spaces allows the manipulation of these properties for improved performance on downstream tasks. One such property is the amount of data spread in a model's…
Multidimensional scaling (MDS) is a family of methods that embed a given set of points into a simple, usually flat, domain. The points are assumed to be sampled from some metric space, and the mapping attempts to preserve the distances…
For a given metric measure space $(X,d,\mu)$ we consider finite samples of points, calculate the matrix of distances between them and then reconstruct the points in some finite-dimensional space using the multidimensional scaling (MDS)…
Neural network weights are typically viewed as the end product of training, while most deep learning research focuses on data, features, and architectures. However, recent advances show that the set of all possible weight values (weight…
Distance metric learning aims to learn from the given training data a valid distance metric, with which the similarity between data samples can be more effectively evaluated for classification. Metric learning is often formulated as a…
We introduce a differential visual similarity metric to train deep neural networks for 3D reconstruction, aimed at improving reconstruction quality. The metric compares two 3D shapes by measuring distances between multi-view images…
We study the problem of supervised learning a metric space under discriminative constraints. Given a universe $X$ and sets ${\cal S}, {\cal D}\subset {X \choose 2}$ of similar and dissimilar pairs, we seek to find a mapping $f:X\to Y$, into…
We initiate the rigorous study of classification in semimetric spaces, which are point sets with a distance function that is non-negative and symmetric, but need not satisfy the triangle inequality. For metric spaces, the doubling dimension…
In this paper, we focus on the model specification problem in multivariate spatial econometric models when a candidate set for the spatial weights matrix is available. We propose a model selection method for the multivariate spatial…
Stationary subspace analysis (SSA) is a blind source separation framework that decomposes linearly mixed multivariate data into stationary and nonstationary components. We extend SSA to spatially indexed data by introducing spatial…
In general, objects can be distinguished on the basis of their features, such as color or shape. In particular, it is assumed that similarity judgments about such features can be processed independently in different metric spaces. However,…