Euclidean Representations Research Articles

Euclidean mirrors and dynamics in network time series

– Analyzing changes in network evolution is central to statistical network inference. We consider a dynamic network model in which each node has an associated time-varying low-dimensional latent vector of feature data, and connection probabilities are functions of these vectors. Under mild assumptions, the evolution of latent vectors exhibits low-dimensional manifold structure under a suitable distance. This distance can be approximated by a measure of separation between the observed networks themselves, and there exist Euclidean representations for underlying network structure, as characterized by this distance. These Euclidean representations, called Euclidean mirrors, permit the visualization of network dynamics and lead to methods for change point and anomaly detection in networks. We illustrate our methodology with real and synthetic data, and identify change points corresponding to massive shifts in pandemic policies in a communication network of a large organization.

The format of the cognitive map depends on the structure of the environment.

Humans and animals form cognitive maps that allow them to navigate through large-scale environments. Here we address a central unresolved question about these maps: whether they exhibit similar characteristics across all environments, or-alternatively-whether different environments yield different types of maps. To investigate this question, we examined spatial learning in three virtual environments: an open courtyard with patios connected by paths (open maze), a set of rooms connected by corridors (closed maze), and a set of isolated rooms connected only by teleporters (teleport maze). All three environments shared the same underlying topological graph structure. Postlearning tests showed that participants formed representations of the three environments that varied in accuracy, format, and individual variability. The open maze was most accurately remembered, followed by the closed maze, and then the teleport maze. In the open maze, most participants developed representations that reflected the Euclidean structure of the space, whereas in the teleport maze, most participants constructed representations that aligned more closely with a mental model of an interconnected graph. In the closed maze, substantial individual variability emerged, with some participants forming Euclidean representations and others forming graph-like representations. These results indicate that an environment's features shape the quality and nature of the spatial representations formed within it, determining whether spatial knowledge takes a Euclidean or graph-like format. Consequently, experimental findings obtained in any single environment may not generalize to others with different features. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Hyperbolic prototypical network for few shot remote sensing scene classification

Recently, in the computer vision and machine learning (ML) communities, a growing interest has been directed to similarity measures operating in hyperbolic spaces due to the geometric properties of these spaces which make them very suitable for embedding data with an underlying hierarchy. These hyperbolic spaces, although increasingly adopted, have received limited attention in the remote sensing (RS) community despite the hierarchical nature of RS data. The objective of this study is therefore to examine the relevance of hyperbolic embeddings of RS data, in particular when addressing the few-shot remote sensing scene classification problem. We adopt hyperbolic prototypical networks as a meta-learning approach to embed scene images along with a feature clipping technique to ensure a more numerically steady model. We then examine whether hyperbolic embeddings provide a better representation than Euclidean representations and better reflect the underlying structure of scene classes. Experimental results on the NWPU-RESISC45 RS dataset demonstrate the superiority of hyperbolic embeddings over their Euclidean counterparts. Our study provides a new perspective by suggesting that operating in hyperbolic spaces is an interesting alternative for the RS community.

Human spatial learning strategies in wormhole virtual environments

ABSTRACT Humans can learn spatial information through navigation in the environment. The nature of these spatial representations is constantly debated, including whether they conform to Euclidean geometry. The present study examined the types of Euclidean representations people may form while learning virtual wormhole mazes. Participants explored Euclidean or non-Euclidean tunnel mazes and drew maps of the landmark layout on a 2D canvas. The results showed that people have different, consistent strategies, some mainly preserving distance information while others mainly preserving turning angles. The straightness of the segments was mostly preserved. These results suggest that representations of non-Euclidean space may be highly variable across individuals, and possible Euclidean solutions need to be carefully examined before testing Euclidean vs alternative models.

Incidence structures near configurations of type (n_3)

An ( n 3 ) configuration is an incidence structure equivalent to a linear hypergraph on n vertices which is both 3-regular and 3-uniform. We investigate a variant in which one constraint, say 3-regularity, is present, and we allow exactly one line to have size four, exactly one line to have size two, and all other lines to have size three. In particular, we study planar (Euclidean or projective) representations, settling the existence question and adapting Steinitz’ theorem for this setting.

Monte Carlo Geometry Modeling for Particle Transport Using Conformal Geometric Algebra

The Monte Carlo (MC) simulations are considered the gold-standard method for calculating the transport and interaction of radiation with the matter. A fundamental component of any MC simulation is the geometrical modeling. Current implementations of the geometrical modeling are based only on the Euclidean representations. However, Euclidean representations may not be the best option for speed up the geometric debugging-modeling computations of radiation transport, due to the number of operations involved in the estimation of position and direction of particles within each geometry shape. In this work, it is proposed for the first time, the use of Conformal Geometric Algebra (CGA), for geometric modeling in MC simulation for radiation transport. In this context, we present some elemental CGA equations for the microscopic modeling of positions and rotations of a radiation particle and the macroscopic modeling of geometrical shapes. It is shown that it is possible to take advantage of the expression power of CGA to create and debug geometry modeling with a triboelectric X-ray application. Additionally, some advantages of the CGA for the microscopic geometric computations are explored.

On a problem of Specker about Euclidean representations of finite graphs

Say that a graph G is representable inRn if there is a map f from its vertex set into the Euclidean space Rn such that ‖f(x)−f(x′)‖=‖f(y)−f(y′)‖ iff {x,x′} and{y,y′} are both edges or both non-edges in G. The purpose of this note is to present the proof of the following result, due to Einhorn and Schoenberg in Einhorn and Schoenberg (1966): if G finite is neither complete nor independent, then it is representable in R|G|−2. A similar result also holds in the case of finite complete edge-colored graphs.

Metrics in Keplerian orbits quotient spaces

Quotient spaces of Keplerian orbits are important instruments for the modelling of orbit samples of celestial bodies on a large time span. We suppose that variations of the orbital eccentricities, inclinations and semi-major axes remain sufficiently small, while arbitrary perturbations are allowed for the arguments of pericentres or longitudes of the nodes, or both. The distance between orbits or their images in quotient spaces serves as a numerical criterion for such problems of Celestial Mechanics as search for common origin of meteoroid streams, comets, and asteroids, asteroid families identification, and others. In this paper, we consider quotient sets of the non-rectilinear Keplerian orbits space $$\mathbb H$$ . Their elements are identified irrespective of the values of pericentre arguments or node longitudes. We prove that distance functions on the quotient sets, introduced in Kholshevnikov et al. (Mon Not R Astron Soc 462:2275–2283, 2016), satisfy metric space axioms and discuss theoretical and practical importance of this result. Isometric embeddings of the quotient spaces into $$\mathbb R^n$$ , and a space of compact subsets of $$\mathbb H$$ with Hausdorff metric are constructed. The Euclidean representations of the orbits spaces find its applications in a problem of orbit averaging and computational algorithms specific to Euclidean space. We also explore completions of $$\mathbb H$$ and its quotient spaces with respect to corresponding metrics and establish a relation between elements of the extended spaces and rectilinear trajectories. Distance between an orbit and subsets of elliptic and hyperbolic orbits is calculated. This quantity provides an upper bound for the metric value in a problem of close orbits identification. Finally the invariance of the equivalence relations in $$\mathbb H$$ under coordinates change is discussed.

Betweenness, Łukasiewicz Rough Inclusions, Euclidean Representations in Information Systems, Hyper–granules and Conflict Resolution*

The purpose of this note is to augment information systems with a relation of betweenness among things in the universe of the system and derive from it a geometric representation of granules of things in Euclidean spaces. Next, the notion of betweenness renders a service in introducing a new notion of a hyper– granule. An application to conflict resolution by defining coalitions of agents as granules or hyper–granules and their mixed strategies as elements of convex hulls spanned on things defining the granule is proposed. Finally, hyper–granules offer a new classifying algorithm which exploits neighborhoods of things and in a sense is an improved with respect to similarity variant of nearest neighbor classifier.

Convex optimization learning of faithful Euclidean distance representations in nonlinear dimensionality reduction

Classical multidimensional scaling only works well when the noisy distances observed in a high dimensional space can be faithfully represented by Euclidean distances in a low dimensional space. Advanced models such as Maximum Variance Unfolding (MVU) and Minimum Volume Embedding (MVE) use Semi-Definite Programming (SDP) to reconstruct such faithful representations. While those SDP models are capable of producing high quality configuration numerically, they suffer two major drawbacks. One is that there exist no theoretically guaranteed bounds on the quality of the configuration. The other is that they are slow in computation when the data points are beyond moderate size. In this paper, we propose a convex optimization model of Euclidean distance matrices. We establish a non-asymptotic error bound for the random graph model with sub-Gaussian noise, and prove that our model produces a matrix estimator of high accuracy when the order of the uniform sample size is roughly the degree of freedom of a low-rank matrix up to a logarithmic factor. Our results partially explain why MVU and MVE often work well. Moreover, the convex optimization model can be efficiently solved by a recently proposed 3-block alternating direction method of multipliers. Numerical experiments show that the model can produce configurations of high quality on large data points that the SDP approach would struggle to cope with.

Action Recognition Using Rate-Invariant Analysis of Skeletal Shape Trajectories.

We study the problem of classifying actions of human subjects using depth movies generated by Kinect or other depth sensors. Representing human body as dynamical skeletons, we study the evolution of their (skeletons’) shapes as trajectories on Kendall’s shape manifold. The action data is typically corrupted by large variability in execution rates within and across subjects and, thus, causing major problems in statistical analyses. To address that issue, we adopt a recently-developed framework of Su et al. [1], [2] to this problem domain. Here, the variable execution rates correspond to re-parameterizations of trajectories, and one uses a parameterization-invariant metric for aligning, comparing, averaging, and modeling trajectories. This is based on a combination of transported square-root vector fields (TSRVFs) of trajectories and the standard Euclidean norm, that allows computational efficiency. We develop a comprehensive suite of computational tools for this application domain: smoothing and denoising skeleton trajectories using median filtering, up- and down-sampling actions in time domain, simultaneous temporal-registration of multiple actions, and extracting invertible Euclidean representations of actions. Due to invertibility these Euclidean representations allow both discriminative and generative models for statistical analysis. For instance, they can be used in a SVM-based classification of original actions, as demonstrated here using MSR Action-3D, MSR Daily Activity and 3D Action Pairs datasets. Using only the skeletal information, we achieve state-of-the-art classification results on these datasets.

Symmetry, oriented matroids and two conjectures of Michel Las Vergnas

The paper has two parts. In the first part we survey the existing results on the cube conjecture of Las Vergnas. This conjecture claims that the orientation of the matroid of the cube is determined by the symmetries of the underlying matroid. The second part deals with Euclidean representations of matroids as geometric simplicial complexes defined by symmetry properties abstracting those of zonotopes. Both sections involve arguments concerning simplicial regions illustrating, once more, the fundamental importance of the simplex conjecture of Las Vergnas.

Sparse conformal prediction for dissimilarity data

Existing classification algorithms focus on vectorial data given in Euclidean space or representations by means of positive semi-definite kernel matrices. Many real world data, like biological sequences are not vectorial, often non-euclidean and given only in the form of (dis-)similarities between examples, requesting for efficient and interpretable models. Vectorial embeddings or transformations to get a valid kernel are limited and current dissimilarity classifiers often lead to dense complex models which are hard to interpret by domain experts. They also fail to provide additional information about the confidence of the classification. In this paper we propose a prototype-based conformal classifier for dissimilarity data. It is based on a prototype dissimilarity learner and extended by the conformal prediction methodology. It (i) can deal with dissimilarity data characterized by an arbitrary symmetric dissimilarity matrix, (ii) offers intuitive classification in terms of sparse prototypical class representatives, (iii) leads to state-of-the-art classification results supported by a confidence measure and (iv) the model complexity is automatically adjusted. In experiments on dissimilarity data we investigate the effectiveness with respect to accuracy and model complexity in comparison to different state of the art classifiers.

A geometrical characterization of strongly regular graphs

We give a necessary and sufficient condition of a Euclidean representation of a simple graph to be spherical. Moreover we show a characterization of strongly regular graphs from the view point of Euclidean representations of a graph. From this characterization, we define a natural generalized concept of a strongly regular graph closely related to Euclidean designs and codes.

Minimal Euclidean representations of graphs

A simple graph G is representable in a real vector space of dimension m , if there is an embedding of the vertex set in the vector space such that the Euclidean distance between any two distinct vertices is one of only two distinct values, α and β , with distance α if the vertices are adjacent and distance β otherwise. The Euclidean representation number of G is the smallest dimension in which G is representable. In this note, we bound the Euclidean representation number of a graph using multiplicities of the eigenvalues of the adjacency matrix. We also give an exact formula for the Euclidean representation number using the main angles of the graph.

Removal and contraction operations to define combinatorial pyramids: application to the design of a spatial modeler

Removal and contraction are basic operations for several methods conceived in order to handle irregular image pyramids, for multi-level image analysis for instance. We give the definitions of removal and contraction operations in the generalized maps framework. We propose a first experimentation of irregular pyramid as a basis for a discrete geometrical modeler that can handle both discrete and continuous representations of geometrical objects. This modeler is based on a pyramidal kernel with four coexisting level between the discrete and the Euclidean representations. We describe how this pyramid can be constructed and updated.

A Comparison of the Multidimensional Scaling of Triadic and Dyadic Distances

We examine the use of triadic distances as a basis for multidimensional scaling (MDS). The MDS of triadic distances (MDS3) and a conventional MDS of dyadic distances (MDS2) both give Euclidean representations. Our analysis suggests that MDS2 and MDS3 can be expected to give very similar results, and this is strongly supported by numerical examples. We have concentrated on the perimeter and generalized Euclidean models of triadic distances, both of which are linear transformations of dyadic distances and so might be suspected of explaining our findings; however an MDS3 of the nonlinear variance definition of triadic distance also closely approximated the MDS2 representation. An appendix gives some matrix results that we have found useful and also gives matrix respresentations and alternative derivations of some known properties of triadic distances.

The Euclidean Representations of the Fibonacci Groups

The Euclidean Representations of the Fibonacci Groups

Euclidean Metric Representations of Haptically Explored Triangles

This study explored whether people create Euclidean representations of 2-dimensional right triangles from touch and use them to make spatial inferences in accord with Euclidean distance axioms. Blindfolded participants who were instructed to form visual images of triangles felt the vertical and horizontal sides of right triangles, then estimated the lengths (but not the angles) of the 3 triangle sides. In these 3 experiments, length estimates conformed closely to the Euclidean metric when evaluated on application of the Pythagorean theorem. Participants who used a visual imaging strategy were accurate more often than those who used visual imagery less often. In Experiments 2 and 3, a hypotenuse inference was as accurate as a direct haptic judgment of the hypotenuse. These results demonstrated similar accuracy of the hypotenuse judgments when participants made verbal rather than haptic estimates. The findings indicate that participants can form Euclidean representations under certain conditions from felt 2-dimensional right triangles based on visual images.

Unitary and Euclidean representations of a quiver

A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (Euclidean) vector space and to each arrow a linear mapping of the corresponding vector spaces. We recall an algorithm for reducing the matrices of a unitary representation to canonical form, give a certain description of the representations of canonical form, and reduce the problem of classifying Euclidean representations to the problem of classifying unitary representations. We also describe the set of dimensions of all indecomposable unitary (Euclidean) representations of a quiver and establish the number of parameters in an indecomposable unitary representation of a given dimension.