Global Geometric Properties Research Articles

Metric geometry of spaces of persistence diagrams

Persistence diagrams are objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence diagrams. In order to do this, we construct a family of functors Dp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {D}}_p$$\\end{document}, 1≤p≤∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le p \\le \\infty $$\\end{document}, that assign, to each metric pair (X, A), a pointed metric space Dp(X,A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {D}}_p(X,A)$$\\end{document}. Moreover, we show that D∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {D}}_{\\infty }$$\\end{document} is sequentially continuous with respect to the Gromov–Hausdorff convergence of metric pairs, and we prove that Dp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {D}}_p$$\\end{document} preserves several useful metric properties, such as completeness and separability, for p∈[1,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\in [1,\\infty )$$\\end{document}, and geodesicity and non-negative curvature in the sense of Alexandrov, for p=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=2$$\\end{document}. For the latter case, we describe the metric of the space of directions at the empty diagram. We also show that the Fréchet mean set of a Borel probability measure on Dp(X,A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {D}}_p(X,A)$$\\end{document}, 1≤p≤∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le p \\le \\infty $$\\end{document}, with finite second moment and compact support is non-empty. As an application of our geometric framework, we prove that the space of Euclidean persistence diagrams, Dp(R2n,Δn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {D}}_{{p}}({\\mathbb {R}}^{2n},\\Delta _n)$$\\end{document}, 1≤n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le n$$\\end{document} and 1≤p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le p<\\infty $$\\end{document}, has infinite covering, Hausdorff, asymptotic, Assouad, and Assouad–Nagata dimensions.

Invisibility enables super-visibility in electromagnetic imaging

This paper is concerned with the inverse electromagnetic scattering problem for anisotropic media. We use the interior resonant modes to develop an inverse scattering scheme for imaging the scatterer. The whole procedure consists of three phases. First, we determine the interior Maxwell transmission eigenvalues of the scatterer from a family of far-field data by the mechanism of the linear sampling method. Next, we determine the corresponding transmission eigenfunctions by solving a constrained optimization problem. Finally, based on both global and local geometric properties of the transmission eigenfunctions, we design an imaging functional which can be used to determine the shape of the medium scatterer. We provide rigorous theoretical basis for our method. Numerical experiments verify the effectiveness, better accuracy and super-resolution results of the proposed scheme.

Geometric Inductive Biases for Identifiable Unsupervised Learning of Disentangled Representations

The model identifiability is a considerable issue in the unsupervised learning of disentangled representations. The PCA inductive biases revealed recently for unsupervised disentangling in VAE-based models are shown to improve local alignment of latent dimensions with principal components of the data. In this paper, in additional to the PCA inductive biases, we propose novel geometric inductive biases from the manifold perspective for unsupervised disentangling, which induce the model to capture the global geometric properties of the data manifold with guaranteed model identifiability. We also propose a Geometric Disentangling Regularized AutoEncoder (GDRAE) that combines the PCA and the proposed geometric inductive biases in one unified framework. The experimental results show the usefulness of the geometric inductive biases in unsupervised disentangling and the effectiveness of our GDRAE in capturing the geometric inductive biases.

Robust measurement of wave function topology on NISQ quantum computers

Topological quantum phases of quantum materials are defined through their topological invariants. These topological invariants are quantities that characterize the global geometrical properties of the quantum wave functions and thus are immune to local noise. Here, we present a strategy to measure topological invariants on quantum computers. We show that our strategy can be easily integrated with the variational quantum eigensolver (VQE) so that the topological properties of generic quantum many-body states can be characterized on current quantum hardware. We demonstrate the robust nature of the method by measuring topological invariants for both non-interacting and interacting models, and map out interacting quantum phase diagrams on quantum simulators and IBM quantum hardware.

The global bifurcation and geometric properties of steady periodic equatorial internal waves with fixed-depth

In this paper, we mainly consider steady periodic equatorial internal waves with a fixed mean depth d>0. By extending the local bifurcation curve obtained in [1], we prove the existence of equatorial internal waves of large amplitude, where the analytic bifurcation theorem and analysis of nodal patterns play a key role. Furthermore, we also obtain some important geometric properties on equatorial internal waves, including the concavity and convexity and symmetry of wave profile.

Reconstruction of Artifacts from Digital Image Repositories

The U.S. Patent Office maintains an archive of cultural artifacts of both ornamental and functional designs. Design patents protect “any new, original, and ornamental design for an article of manufacture” [ 23 ] such as busts, statues, and the shape or configuration of any article of manufacture. The Patent Office archive originally contained both drawings and models as an effort to preserve a chronological history of the science and culture of the United States. This collection was devastated by two fires in 1836 and 1877, which destroyed or damaged more than 100,000 patent models. The Patent Office began divesting its collection in 1908, and Congress ultimately chose to dissolve the collection in 1926. A total of 10,000 of the original models are preserved in the collections of the Smithsonian National Museum of American History in Washington, D.C., and another 5,000 are preserved in the collections of the Hagley Museum and Library in Wilmington, Delaware. Although most of the physical artifacts no longer exist, documents describing the original physical models remain. These documents show each artifact as a collection of multiple black and white images depicting the artifact from different views. We propose an algorithm to reconstruct 3D point cloud models of the original artifacts from multiview images collected directly from the Patent Office archives. The use of 3D point cloud models allows both local and global geometric properties of the artifact to be described in a single vector representation. This vector representation of the artifact may be used individually or in combination with other descriptors for retrieval and classification tasks. For collections that are unclassified or where differing classification systems are used, these vector representations allow clustering of artifacts into meaningful groupings.

On the Svelteness as an Engineering Tool in Constructal Design: A Critical Review

The application of Constructal theory to the flow design in engineering applications connects the channels’ architecture with their freedom to morph. Assessing the evolution of the flow architecture in Constructal Design requires a core parameter. Svelteness is the best candidate, given its definition as a flow architecture’s intrinsic global geometric property. However, despite the broad applicability range of Constructal theory, research has restricted the use of Svelteness to fluid flow, focusing on using it to justify disregarding local pressure losses compared to distributed friction losses, connecting the design of the flow to its survival. This work reviews the application of Svelteness, from the intuitive perception of its meaning to its use in engineering design, namely understanding the difference between assuming the impact of Svelteness versus considering the effects of its evolution in time. This understanding allows exploring the depth and validity of applying Svelteness as a universal criterion, comparing the different methods that define it, and discussing its relevance to explaining freedom to morph in a flow. Using two types of configurations (serpentine and canopy-to-canopy), the review shows the relevance of using the configuration area for the external length scale in the presence of ramifications and a relation between the configuration area and the path followed by what flows in the absence of configurations. Finally, we discuss the establishment of Svelteness as an engineering design tool using the law of diminishing returns.

Localized Shape Modelling with Global Coherence: An Inverse Spectral Approach

AbstractMany natural shapes have most of their characterizing features concentrated over a few regions in space. For example, humans and animals have distinctive head shapes, while inorganic objects like chairs and airplanes are made of well‐localized functional parts with specific geometric features. Often, these features are strongly correlated – a modification of facial traits in a quadruped should induce changes to the body structure. However, in shape modelling applications, these types of edits are among the hardest ones; they require high precision, but also a global awareness of the entire shape. Even in the deep learning era, obtaining manipulable representations that satisfy such requirements is an open problem posing significant constraints. In this work, we address this problem by defining a data‐driven model upon a family of linear operators (variants of the mesh Laplacian), whose spectra capture global and local geometric properties of the shape at hand. Modifications to these spectra are translated to semantically valid deformations of the corresponding surface. By explicitly decoupling the global from the local surface features, our pipeline allows to perform local edits while simultaneously maintaining a global stylistic coherence. We empirically demonstrate how our learning‐based model generalizes to shape representations not seen at training time, and we systematically analyze different choices of local operators over diverse shape categories.

Positive and negative triangularity in RFX-mod2: a comparative analysis

We present a comparative analysis of practically achievable positive and negative triangularity configurations in the next RFX-mod2 tokamak campaign. The designed single-null positive triangularity plasmas—based on analogous, formerly realized scenarios in RFX-mod—are mirrored, keeping most of the other parameters fixed. In this procedure, we show how some local and global geometric properties of the plasma are modified, and how these properties reflect on changes in vertical stability, low-n ideal stability and electrostatic turbulence level.

Ancient asymptotically cylindrical flows and applications

In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of the mean curvature flow in $${\mathbb {R}}^{n+1}$$ for all $$n\ge 3$$ : we show that if a mean curvature flow $$\{M_t\}$$ in $${\mathbb {R}}^{n+1}$$ has an $$S^{n-1}\times {\mathbb {R}}$$ singularity at $$(x_0,t_0)$$ , then there exists an $$\varepsilon =\varepsilon (x_0,t_0)>0$$ such that $$M_t\cap B(x_0,\varepsilon )$$ is mean-convex for all $$t\in (t_0-\varepsilon ^2,t_0+\varepsilon ^2)$$ . As in the case $$n=2$$ , which was resolved by the first three authors in Choi et al. (Acta Math, 2018), the existence of such a mean-convex neighborhood follows from classifying a certain class of ancient Brakke flows that arise as potential blowup limits near a neck singularity. Specifically, we prove that any ancient unit-regular integral Brakke flow with a cylindrical blowdown must be either a round shrinking cylinder, a translating bowl soliton, or an ancient oval. In particular, combined with a prior result of the last two authors (Hershkovits and White in Commun Pure Appl Math 73(3):558–580, 2020), we obtain uniqueness of mean curvature flow through neck singularities. The main difficulty in addressing the higher dimensional case is in promoting the spectral analysis on the cylinder to global geometric properties of the solution. Most crucially, due to the potential wide variety of self-shrinking flows with entropy lower than the cylinder when $$n\ge 3$$ , smoothness does not follow from the spectral analysis by soft arguments. This precludes the use of the classical moving plane method to derive symmetry. To overcome this, we introduce a novel variant of the moving plane method, which we call “moving plane method without assuming smoothness”—where smoothness and symmetry are established in tandem.

Emergent probability fluxes in confined microbial navigation

When the motion of a motile cell is observed closely, it appears erratic, and yet the combination of nonequilibrium forces and surfaces can produce striking examples of organization in microbial systems. While most of our current understanding is based on bulk systems or idealized geometries, it remains elusive how and at which length scale self-organization emerges in complex geometries. Here, using experiments and analytical and numerical calculations, we study the motion of motile cells under controlled microfluidic conditions and demonstrate that probability flux loops organize active motion, even at the level of a single cell exploring an isolated compartment of nontrivial geometry. By accounting for the interplay of activity and interfacial forces, we find that the boundary's curvature determines the nonequilibrium probability fluxes of the motion. We theoretically predict a universal relation between fluxes and global geometric properties that is directly confirmed by experiments. Our findings open the possibility to decipher the most probable trajectories of motile cells and may enable the design of geometries guiding their time-averaged motion.

Geometry of Adjoint-Invariant Submanifolds of SE(3)

This article aims to extend the theory of Lie subgroups and symmetric subspaces for studying an important class of submanifolds of the special Euclidean group SE(3) whose tangent space at each point on the submanifold relates to that at the identity by an adjoint map. These submanifolds, which we call adjoint-invariant submanifolds in this article, are known in the literature as persistent submanifolds, since they are strictly related to the concept of persistent screw systems. The difference is that in this article, just as Lie subgroups and symmetric subspaces, we put forward adjoint-invariant submanifolds as independent geometric objects from mechanisms and their associated local screw systems. Adjoint invariance relaxes the strict left and right invariance of Lie subgroups and the reflective invariance of symmetric subspaces by allowing generic moving reference frame in the aforementioned adjoint map. It turns out such adjoint invariance can be studied under the framework of distributions on manifolds, which allows us to explore global geometric properties of adjoint-invariant submanifolds. We classify adjoint-invariant submanifolds into reflective-type and product-type submanifolds and derive the conditions for their adjoint invariance. We then propose geometric methods and algorithms for synthesizing the kinematic generators for reflective-type submanifolds, as demonstrated with a number of examples.

A note on causality conditions on covering spacetimes

A number of techniques in Lorentzian geometry, such as those used in the proofs of singularity theorems, depend on certain smooth coverings retaining interesting global geometric properties, including causal ones. In this note we give explicit examples showing that, unlike some of the more commonly adopted rungs of the causal ladder such as strong causality or global hyperbolicity, less-utilized conditions such as causal continuity or causal simplicity do not in general pass to coverings, as already speculated by one of the authors (EM). As a consequence, any result which relies on these causality conditions transferring to coverings must be revised accordingly. In particular, some amendments in the statement and proof of a version of the Gannon–Lee singularity theorem previously given by one of us (IPCS) are also presented here that address a gap in its original proof, simultaneously expanding its scope to spacetimes with lower causality.

Kinematics of coupler curves for spherical four-bar linkages based on new spherical adjoint approach

The local and global geometric properties of spherical coupler curves constitute spherical kinematics of spherical four-bar linkages, which can be adopted to reveal distribution characteristics of spherical coupler curves. New unified spherical adjoint approach is established in the paper to study both the local and global geometric properties in order to enrich the atlas of spherical coupler curves with geometric characteristics. Since the constraint curve of spherical four-bar linkage is a simple spherical circle and the spherical centrodes imply intrinsic properties of spherical motion of the coupler link, they are in their turn taken as the original curves in spherical adjoint approach to derive the geodesic curvature and analyze the local geometric characteristics of the spherical coupler curves. The conditions for different spherical double points, such as spherical crunodes, tacnodes and cusps of the spherical coupler curve are derived through the spherical adjoint approach. The spherical surface of the coupler link can be divided into several areas by the spherical moving centrode and the spherical tacnode's tracer curve. The points in each area trace spherical coupler curves with a specific shape. The characteristic points, which trace spherical coupler curves with cusp, geodesic inflection point, spherical Ball point, spherical Burmester point, crunode and tacnode can be readily located in the coupler link by the modelling procedure and the derived condition equations. In the end the distribution of spherical coupler curves with both local and global characteristics is elaborated. The research proposes systematic geometric properties of spherical coupler curves based on the new established approach, and provides a solid theoretical basis for the kinematic analysis and synthesis of the spherical four-bar linkages.

Complexity of operators generated by quantum mechanical Hamiltonians

We propose how to compute the complexity of operators generated by Hamiltonians in quantum field theory (QFT) and quantum mechanics (QM). The Hamiltonians in QFT/QM and quantum circuit have a few essential differences, for which we introduce new principles and methods for complexity. We show that the complexity geometry corresponding to one-dimensional quadratic Hamiltonians is equivalent to AdS3 spacetime. Here, the requirement that the complexity is nonnegative corresponds to the fact that the Hamiltonian is lower bounded and the speed of a particle is not superluminal. Our proposal proves the complexity of the operator generated by a free Hamiltonian is zero, as expected. By studying a non-relativistic particle in compact Riemannian manifolds we find the complexity is given by the global geometric property of the space. In particular, we show that in low energy limit the critical spacetime dimension to ensure the ‘nonnegative’ complexity is the 3+1 dimension.

Hyperbolic and Parabolic Unimodular Random Maps

We show that for infinite planar unimodular random rooted maps. many global geometric and probabilistic properties are equivalent, and are determined by a natural, local notion of average curvature. This dichotomy includes properties relating to amenability, conformal geometry, random walks, uniform and minimal spanning forests, and Bernoulli bond percolation. We also prove that every simply connected unimodular random rooted map is sofic, that is, a Benjamini–Schramm limit of finite maps.

D2 shape distribution and artificial neural networks for 3D objects recognition

In this paper, we propose a 3D object recognition approach, based on the shape distribution D2 and artificial neural networks. The challenge is to discriminate between similar and dissimilar shapes by finding a shape signature that can be constructed and classified quickly. We propose a connectionist system to recognize 3D objects in VRML (Virtual Reality Modeling Language) format. The key idea is to represent the signature of an object as a shape distribution sampled from a shape function measuring global geometric properties of an object. The proposed strategy is the following: from a polygon object to be recognized, a triangulation is performed. Then, distances are calculated between two random points of the triangulated surface of the 3D object. The frequency of these distances will be represented by a normalized histogram. The values of these histograms feed a multi-layer neural network with back- propagation training. We demonstrate the potential of this approach in a set of experiments, which proved our system could achieve above 91.7% recognition rate. In addition, to evaluate the efficiency of our method, we compare our classifier with Support vector machine and k- nearest neighbours. The simulation results highlight the performance of the proposed approach.

Extract feature curves on noisy triangular meshes

Feature curve is a powerful shape descriptor for a wide spectrum of applications, such as shape registration, geometry compression and surface reconstruction in CAD and reverse engineering. However, feature curve extraction from noisy data has been a challenging problem for decades. In this paper, we propose a robust algorithm for extracting feature curves from noisy triangular meshes. Specifically, we first slice the triangular mesh into a series of sections along a given slicing guide line interactively stroked by users. Under different levels of noise frequencies and amplitudes, we measure the similarities of adjacent sections and extract their corresponding salient points, followed by connecting the consistent salient points to form feature curves. Instead of exploiting the local geometric information as previous methods do, we leverage the global geometric properties of section curve pairs of the input mesh to extract feature curves. This enables our method to cope with a high level of noise. Moreover, our method automatically computes optimal noise frequencies and amplitudes, which makes feature curve extraction less sensitive to different levels of noise than others. A variety of experiments demonstrate the robustness and effectiveness of our proposed method on low-quality mesh data.

Symplectic Entropy as a Novel Measure for Complex Systems

Real systems are often complex, nonlinear, and noisy in various fields, including mathematics, natural science, and social science. We present the symplectic entropy (SymEn) measure as well as an analysis method based on SymEn to estimate the nonlinearity of a complex system by analyzing the given time series. The SymEn estimation is a kind of entropy based on symplectic principal component analysis (SPCA), which represents organized but unpredictable behaviors of systems. The key to SPCA is to preserve the global submanifold geometrical properties of the systems through a symplectic transform in the phase space, which is a kind of measure-preserving transform. The ability to preserve the global geometrical characteristics makes SymEn a test statistic for the detection of the nonlinear characteristics in several typical chaotic time series, and the stochastic characteristic in Gaussian white noise. The results are in agreement with findings in the approximate entropy (ApEn), the sample entropy (SampEn), and the fuzzy entropy (FuzzyEn). Moreover, the SymEn method is also used to analyze the nonlinearities of real signals (including the electroencephalogram (EEG) signals for Autism Spectrum Disorder (ASD) and healthy subjects, and the sound and vibration signals for mechanical systems). The results indicate that the SymEn estimation can be taken as a measure for the description of the nonlinear characteristics in the data collected from natural complex systems.

Effects of solar wind speed on the secondary energetic neutral source of the Interstellar Boundary Explorer ribbon

The Interstellar Boundary EXplorer (IBEX) ribbon is an intense energetic neutral atom (ENA) emission feature encircling the sky, spanning energies ≤0.5–6 keV. The ribbon may be produced by the “secondary ENA” mechanism, where ENAs emitted from a source plasma population inside the heliosphere propagate outside the heliopause, undergo two charge-exchange events, and become secondary ENAs that may be directed back toward Earth and detected by IBEX. In this scenario, the source plasma population is governed by the interaction of the solar wind (SW) with the interstellar medium and is thus sensitive to the global SW properties. Moreover, this scenario predicts that the distance to the source of secondary ENAs depends on the ENA energy and SW speed, which in turn may affect the shape of the ribbon. In this paper, we use a computational model of the heliosphere with simplified SW boundary conditions to analyze the influence of ENA energy and SW speed, independent of time and latitude, on the global spatial and geometric properties of the ribbon. We find a strong dependence of the simulated ribbon energy spectrum and spatial symmetry on SW speed and ENA energy, and only a slight dependence on ribbon geometry. Our results suggest a significant number of primary ENAs from the inner heliosheath may contribute to the pickup ion source population outside the heliopause, depending on the ENA energy and SW speed. The lack of variation in the simulated ribbon center as a function of ENA energy and SW speed, in contrast to the observations, implies that the asymmetry of the SW plays an important role in determining the position of the ribbon. Comparisons to the IBEX data also signify the ribbon’s dependence on the properties of the local interstellar medium, particularly the interstellar magnetic field.