Hyperbolic Representation Research Articles

Navigating Memorability Landscapes: Hyperbolic Geometry Reveals Hierarchical Structures in Object Concept Memory.

Why are some object concepts (e.g., birds, cars, vegetables, etc.) more memorable than others? Prior studies have suggested that features (e.g., color, animacy, etc.) and typicality (e.g., robin vs. penguin) of object images influences the likelihood of being remembered. However, a complete understanding of object memorability remains elusive. In this study, we examine whether the geometric relationship between object concepts explains differences in their memorability. Specifically, we hypothesize that image concepts will be geometrically arranged in hierarchical structures and that memorability will be explained by a concept's depth in these hierarchical trees. To test this hypothesis, we construct a Hyperbolic representation space of object concepts (N=1,854) from the THINGS database (Hebart et al., 2019), which consists of naturalistic images of concrete objects, and a space of 49 feature dimensions derived from data-driven models. Using ALBATROSS (Stier, A. J., Giusti, C., & Berman, M. G., In prep), a stochastic topological data analysis technique that detects underlying structures of data, we demonstrate that Hyperbolic geometry efficiently captures the hierarchical organization of object concepts above and beyond a traditional Euclidean geometry and that hierarchical organization is related to memorability. We find that concepts closer to the center of the representational space are more prototypical and also more memorable. Importantly, Hyperbolic distances are more predictive of memorability and prototypicality than Euclidean distances, suggesting that concept memorability and typicality are organized hierarchically. Taken together, our work presents a novel hierarchical representational structure of object concepts that explains memorability and typicality.

Simulacrum And Hyperreality in Cargill’s Day Zero

This study examines the simulacrum and simulation representation of postmodern society in Cargill's Day Zero in the postmodern theoretical paradigm of Baudrillard. The postmodern concept of hyperreality is concerned with concealing the truth behind reproductions that is twice removed from reality. The thorough study of the text shows how simulacra and simulation can produce highly real environments that seem to exist entirely outside of human reality. This writing is dystopian in nature because the hyperbolic representation of consumer society leads towards magic world which seems real due to its continuous practice. The protagonist of the story is a little boy named Ezra who uses a robot named Pounce to help him to navigate the post-apocalyptic world. Pounce, a fictitious nannybot, is the representation of consumer society due to its robotic implications. Pounce wonders over the elements of robot revolution that will exterminate humanity. This paper has analyzed the rebellious attitude of machinery such as by robots and other artificially intelligent computers against humans. Therefore, the ordinary and uninteresting state of the modern world eventually gives rise to the world of hyperreality, where everything appears to be more fascinating and less dry than the alleged desert of the real outside. In the context of this study, it is asserted that the postmodern person gets tired of the apparent artificiality of hyperreality and either strives to re-discover the real or confronts the meaningless wasteland left in the wake of the real’s absence. Through this analysis, the researcher has drawn the conclusion that reality has completely changed and that hyperreality is now the predominate reality. Therefore, the application of this study is valuable in demonstrating an understanding of the scope and impact of hyperreality, simulation, and simulacra on the postmodern individuals.

A Construction Method of Hyperbolic Representation Lexicon Oriented to Chinese Ironic Text

A Construction Method of Hyperbolic Representation Lexicon Oriented to Chinese Ironic Text

DHGAT: Hyperbolic representation learning on dynamic graphs via attention networks

Hyperbolic graph embedding has garnered significant attention in a wide range of downstream applications. This is because hyperbolic geometry provides a valuable mapping tool, whereby the scale-free or hierarchical properties of complex networks can be naturally reflected as hyperbolic metric properties. Despite this, most advancements have focused on learning hyperbolic representations of static graphs, the potential advantages of hyperbolic metrics in dynamic graphs embedding have not been fully exploited. To fully harness the properties of hyperbolic space, we propose DHGAT11https://github.com/Lihaogx/DHGAT., a dynamic hyperbolic graph attention network with a novel architecture that designs a spatiotemporal self-attention mechanism based on hyperbolic distance. DHGAT maps dynamic graphs into the hyperbolic space, generates spatiotemporal self-attention of nodes, and directly aggregates weighted node representations without the tangent space. This is achieved by using the Einstein gyromidpoints, which reduces distortions and preserves manifold properties. The results of experiments on real-world datasets demonstrate that DHGAT performs exceptionally well in multi-step link prediction tasks, particularly in predicting new links compared to the baselines.

Knowledge graph completion method based on hyperbolic representation learning and contrastive learning

Knowledge graph completion employs existing triples to deduce missing data, thereby enriching and enhancing graph completeness. Recent research has revealed that using hyperbolic representation learning in knowledge graph completion yields superior expressive and generalization capabilities. However, the long-tail problem and the presence of hyperbolic metrics make it challenging to effectively learn low-frequency entities or relations, resulting in embedding space distortion and impacting the original semantic relationships. Therefore, this paper proposes a knowledge graph completion method (Att-CL) that integrates hyperbolic representation learning and contrastive learning. In this approach, knowledge is embedded into a hyperbolic space, and samples with limited hierarchical characteristics and insufficient feature information are enhanced by introducing adversarial noise. The loss function of the embedded samples is backpropagated into embedding vectors, perturbations are adjusted in the gradient direction to promote smoothness and locality, and hyperparameters are introduced for fine-tuning the adversarial strength in the construction of adversarial samples for data augmentation to enhance model robustness. To mitigate data distortion due to hyperbolic metrics, a penalty term is introduced in the contrastive loss function to control the distances of the embedding vectors from the origin, thereby reducing the impact of the metrics and further improving the model's completion ability. Experimental results on the WN18RR and FB15K-237 benchmark datasets demonstrate significant improvements in metrics such as MRR, Hits@1, and Hits@3 compared to traditional knowledge graph completion models, providing ample evidence of the model's effectiveness.

Quantum exponentials for the modular double and applications in gravity models

In this note, we propose a decomposition of the quantum matrix group SLq+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\ extrm{SL}}_q^{+} $$\\end{document}(2, ℝ) as (deformed) exponentiation of the quantum algebra generators of Faddeev’s modular double of Uq(sl\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{sl} $$\\end{document}(2, ℝ)). The formula is checked by relating hyperbolic representation matrices with the Whittaker function. We interpret (or derive) it in terms of Hopf duality, and use it to explicitly construct the regular representation of the modular double, leading to the Casimir and its modular dual as the analogue of the Laplacian on the quantum group manifold. This description is important for both 2d Liouville gravity, and 3d pure gravity, since both are governed by this algebraic structure. This result builds towards a q-BF formulation of the amplitudes of both of these gravitational models.

Hyperbolic Neural Collaborative Recommender

Recently, deep learning techniques have yielded immense success on recommender systems. However, one weakness of most deep methods is that, users/items mutual semantic relationships, which are latent in the user-item interactions, are not distilled out explicitly. Moreover, most methods have been primarily focused on representation learning in Euclidean geometry. Since recent studies have shown that the bipartite graph structure has the non-Euclidean latent anatomy, Euclidean embeddings may suffer from a certain degree of distortion. In this work, we present Hyperbolic Neural Collaborative Recommender (HNCR), a deep hyperbolic representation learning method that exploits mutual semantic relationships among users/items for collaborative filtering tasks. HNCR first introduces a neighbor construction strategy to build user and item semantic neighborhoods. Then HNCR develops a framework based on deep learning and hyperbolic geometry to integrate constructed neighborhoods into recommendation. To evaluate our method, we conduct experiments on the four datasets. Experimental results show the superiority of HNCR compared with its Euclidean counterpart and state-of-the-art recommendation baselines. The results also indicate that hyperbolic representations can reflect meaningful data insights.

Hyperbolic odorant mixtures as a basis for more efficient signaling between flowering plants and bees.

Animals use odors in many natural contexts, for example, for finding mates or food, or signaling danger. Most analyses of natural odors search for either the most meaningful components of a natural odor mixture, or they use linear metrics to analyze the mixture compositions. However, we have recently shown that the physical space for complex mixtures is ‘hyperbolic’, meaning that there are certain combinations of variables that have a disproportionately large impact on perception and that these variables have specific interpretations in terms of metabolic processes taking place inside the flower and fruit that produce the odors. Here we show that the statistics of odorants and odorant mixtures produced by inflorescences (Brassica rapa) are also better described with a hyperbolic rather than a linear metric, and that combinations of odorants in the hyperbolic space are better predictors of the nectar and pollen resources sought by bee pollinators than the standard Euclidian combinations. We also show that honey bee and bumble bee antennae can detect most components of the B. rapa odor space that we tested, and the strength of responses correlates with positions of odorants in the hyperbolic space. In sum, a hyperbolic representation can be used to guide investigation of how information is represented at different levels of processing in the CNS.

Hyperbolic Representation of Lateral Force–Displacement Relationship for Underground Installed Pipe

Extreme natural hazards such as earthquakes, landslides, and liquefaction create permanent ground deformation (PGD). With the recognition that PGD often causes the most serious local damage in underground structures such as buried pipelines and mining facilities, research and engineering practices for underground structures have focused on soil–structure interaction under PGD. In this study, an underground pipeline was investigated due to its simple geometry. Geotechnical data collection and analysis were used as a study method. Of key importance is the soil–pipe interaction with respect to PGD below the subsurface. This response is typically highlighted by a force vs. displacement relationship and is primarily a function of soil unit weight, depth from surface to the pipe centerline, and the pipe diameter. The non-linear force vs. displacement relationship for transverse horizontal force on a pipe subjected to lateral ground movement, can be represented by a hyperbola. The nonlinear hyperbola can then be turned into a linear line by transforming the axis. This paper investigates a wide range of soil characteristics and summarizes representative hyperbolic parameters for dry medium, dense, and very dense sand for lateral ground movement. The approach is convenient for modeling the soil–pipe interaction and is critical for addressing the complexities of soil and pipe performance, consistent with real-world soil–pipe behavior. The ideas and data analysis techniques presented in this study may be fine-tuned and applied to more complex problems including mining and could ultimately contribute to the management of geotechnical risks.

Inviscid limit of the compressible Navier–Stokes equations for asymptotically isothermal gases

We prove the existence of a relative finite-energy solution of the one-dimensional, isentropic Euler equations under the assumption of an asymptotically isothermal pressure law, that is, p(ρ)/ρ=O(1) in the limit ρ→∞. This solution is obtained as the vanishing viscosity limit of classical solutions of the one-dimensional, isentropic, compressible Navier–Stokes equations. Our approach relies on the method of compensated compactness to pass to the limit rigorously in the nonlinear terms. Key to our strategy is the derivation of hyperbolic representation formulas for the entropy kernel and related quantities; among others, a special entropy pair used to obtain higher uniform integrability estimates on the approximate solutions. Intricate bounding procedures relying on these representation formulas then yield the required compactness of the entropy dissipation measures. In turn, we prove that the Young measure generated by the classical solutions of the Navier–Stokes equations reduces to a Dirac mass, from which we deduce the required convergence to a solution of the Euler equations.

Age representation of Lévy walks: partial density waves, relaxation and first passage time statistics

Lévy walks (LWs) define a fundamental class of finite velocity stochastic processes that can be introduced as a special case of continuous time random walks. Alternatively, there is a hyperbolic representation of them in terms of partial probability density waves. Using the latter framework we explore the impact of aging on LWs, which can be viewed as a specific initial preparation of the particle ensemble with respect to an age distribution. We show that the hyperbolic age formulation is suitable for a simple integral representation in terms of linear Volterra equations for any initial preparation. On this basis relaxation properties, i.e. the convergence towards equilibrium of a generic thermodynamic function dependent on the spatial particle distribution, and first passage time statistics in bounded domains are studied by connecting the latter problem with solute release kinetics. We find that even normal diffusive LWs, where the long-term mean square displacement increases linearly with time, may display anomalous relaxation properties such as stretched exponential decay. We then discuss the impact of aging on the first passage time statistics of LWs by developing the corresponding Volterra integral representation. As a further natural generalization the concept of LWs with wearing is introduced to account for mobility losses.

The stochastic formulation of the Stephan’s roblem in hyperbolic representation

The stochastic formulation of the Stephan’s roblem in hyperbolic representation

On Hyperbolic Split Quaternions and Hyperbolic Split Quaternion Matrices

The purpose of this study is to investigate some properties of $$2 \times 2$$ hyperbolic split quaternion matrices. To verify this, we use matrices corresponding to the basis of hyperbolic split quaternions. Moreover, we give $$4 \times 4$$ hyperbolic matrix representation and $$8 \times 8$$ real matrix representation of a hyperbolic split quaternion.

On the Concept of Representations of Prime Numbers and Prime Products

The author used to think that the only representation of prime numbers, etc. was the prime number double helix representation. However, in 2011, the author published a paper which puzzled him and it eventually dawned upon him that there was more than one representation of prime numbers. The second representation will be referred to as the hyperbolic representation. The third representation will be called the parabolic representation and is related to the hyperbolic representation. The fourth representation is the triangular representation and is related to the hyperbolic representation. Both of the primary representations, the double helix and the hyperbolic both reject that 2 and 3 are prime numbers. The hyperbolic representation is shown to be related to Lorentz-like transformations.

Logarithmic extension of real numbers and hyperbolic representation of generalized Lorentz transforms

Logarithmic extension of real numbers and hyperbolic representation of generalized Lorentz transforms

Hygroscopic moisture content: determination and correlations

Hygroscopic moisture content of fine-grained soil represents the water adsorbed by the negatively charged clay platelets when the oven-dried soil is exposed to an atmosphere with different levels of humidity, which is a variable. Hygroscopic moisture content has been linked with many clay mineralogical parameters in predictive geotechnical engineering. However, there appears to be no standard codal provisions to quantify this property. This paper uses the modified rectangular hyperbolic representation of the adsorbed water content against time of exposure to an environment with 100% relative humidity relationship to quantify the hygroscopic moisture content. The proposed method is very simple and fairly accurate. The hygroscopic moisture contents so predicted corresponding to 100% relative humidity have been shown to have one-to-one correspondence with the values obtained through actual time-consuming measurements. By virtue of common dependency factors, it is shown that the hygroscopic moisture content at 100% relative humidity of fine-grained soils can be uniquely related with their liquid limits and equilibrium sediment volumes. Furthermore, instead of a time-consuming method of determining the hygroscopic moisture content, a simple method of predicting the same knowing their liquid limits has been proposed.

Hyperbolic model to evaluate uplift force on pile in expansive soils

The piles in expansive soils would experience uplift force on the shaft of the piles due to the swelling of the soils. For a given soil, the magnitude of interface shear stress (uplift force) on pile depends on the interface friction between the pile material and the soil, normal stress acting on the pile surface and the magnitude of swelling, which in turn creates the relative movement at the interface between the soil and the pile material. The hyperbolic representation of stress strain curves has been found to be a convenient and useful means of representing the non-linearity of the stress strain curves of soils and forms an important part of the stress strain relationship. In this paper a simplified hyperbolic model is proposed to evaluate the pile uplift in expansive soil. The hyperbolic model is developed based on the model pile uplift tests, interface shear tests and consolidation test results. The interface shear stress is also evaluated using a program PIES (Axial response of Pile In Expansive Soils). Both the analytical studies gave a good correlation of interface shear stress profile obtained by laboratory studies. Finally a design procedure is proposed to evaluate the uplift force along the shaft of pile in expansive soils.

Deformation of hyperbolic manifolds in 𝑃𝐺𝐿(𝑛,𝐂) and discreteness of the peripheral representations

Let M M be a cusped hyperbolic 3 3 -manifold, e.g. a knot complement. Thurston showed that the space of deformations of its fundamental group in P G L ( 2 , C ) \mathrm {PGL}(2,\mathbf {C}) (up to conjugation) is of complex dimension the number ν \nu of cusps near the hyperbolic representation. It seems natural to ask whether some representations remain discrete after deformation. The answer is generically not. A simple reason for it lies inside the cusps: the degeneracy of the peripheral representation (i.e. representations of fundamental groups of the ν \nu peripheral tori). They indeed generically become non-discrete, except for a countable set. This last set corresponds to hyperbolic Dehn surgeries on M M , for which the peripheral representation is no more faithful. We work here in the framework of P G L ( n , C ) \mathrm {PGL}(n,\mathbf {C}) . The hyperbolic structure lifts, via the n n -dimensional irreducible representation, to a representation ρ g e o m \rho _{\mathrm {geom}} . We know from the work of Menal-Ferrer and Porti that the space of deformations of ρ geom \rho _{\textrm {geom}} has complex dimension ( n − 1 ) ν (n-1)\nu . We prove here that, unlike the P G L ( 2 ) \mathrm {PGL}(2) -case, the generic behaviour becomes the discreteness (and faithfulness) of the peripheral representation: in a neighbourhood of the geometric representation, the non-discrete peripheral representations are contained in a real analytic subvariety of codimension ≥ 1 \geq 1 .

A solvable string on a Lorentzian surface

It is shown that there are nonlinear sigma models which are Darboux integrable and possess a solvable Vessiot group in addition to those whose Vessiot groups are central extensions of semi-simple Lie groups. They govern harmonic maps between Minkowski space R1,1 and certain complete, non-constant curvature 2-metrics. The solvability of the Vessiot group permits a reduction of the general Cauchy problem to quadrature. We treat the specific case of harmonic maps from Minkowski space into a non-constant curvature Lorentzian 2-metric, λ. Despite the completeness of λ we exhibit a Cauchy problem with real analytic initial data which blows up in finite time. We also derive a hyperbolic Weierstrass representation formula for all harmonic maps from R1,1 into λ.

The a-Curve and Personally Patterned Variation: More Tools for Investigating Language Variation and Change

This article explores two variables that largely have been ignored in studies of language variation and language change: frequency and individual speaker. In doing so, it demonstrates the usefulness of the A-curve, an asymptotic hyperbolic graphic representation of language variation based on usage. Data are drawn from varieties of English: two Afro-Caribbean vernaculars (Antiguan and Negerhollands) and East Sutherland Gaelic. Because both Negerhollands and East Sutherland Gaelic were moribund when the data discussed here were collected and both are characterized by personally patterned variation, the histories of these varieties are briefly considered. The article concludes that both frequency and individual speaker are essential for a full understanding of the ways in which language users deploy their linguistic resources and, thus, are critical for our understanding of human language.