Space Of Metrics Research Articles

Metric flows with neural networks

Abstract We develop a general theory of flows in the space of Riemannian metrics induced by neural network (NN) gradient descent. This is motivated in part by recent advances in approximating Calabi–Yau metrics with NNs and is enabled by recent advances in understanding flows in the space of NNs. We derive the corresponding metric flow equations, which are governed by a metric neural tangent kernel (NTK), a complicated, non-local object that evolves in time. However, many architectures admit an infinite-width limit in which the kernel becomes fixed and the dynamics simplify. Additional assumptions can induce locality in the flow, which allows for the realization of Perelman’s formulation of Ricci flow that was used to resolve the 3d Poincaré conjecture. We demonstrate that such fixed kernel regimes lead to poor learning of numerical Calabi–Yau metrics, as is expected since the associated NNs do not learn features. Conversely, we demonstrate that well-learned numerical metrics at finite-width exhibit an evolving metric-NTK, associated with feature learning. Our theory of NN metric flows therefore explains why NNs are better at learning Calabi–Yau metrics than fixed kernel methods, such as the Ricci flow.

Variational principles for Feldman–Katok metric mean dimension

We introduce the notion of Feldman–Katok metric mean dimensions in this note. We show metric mean dimensions defined by different dynamical metrics of phase space coincide under the (weak) tame growth of covering numbers, and establish variational principles for Feldman–Katok metric mean dimensions in terms of FK Katok ϵ-entropy and FK local ϵ-entropy function.

A note on the Cotton flow and the Ricci flow for three-manifolds, and the Hořava–Lifshitz gravity

We consider the more general geometrical flow in the space of metrics for three-manifolds that consists of a combination of two flows, the Cotton flow and the Ricci flow; by playing a fundamental role in the detailed balance principle of the four dimensional Hořava–Lifshitz gravity, this generalized flow reveals another difficulty with this theory that attempts to be a candidate for an UV completion of Einstein general relativity, namely, the supposed emergency of the speed of light, the Newton constant, and the cosmological constant, from a deeply nonrelativistic theory of gravity. Respecting that principle, the generalized flow shows the proliferation of different limits of the theory with an unwanted behavior at both the IR and UV regimes.

Quantum geometrodynamics revived: II. Hilbert space of positive definite metrics

Abstract This paper represents the second in a series of works aimed at reinvigorating the quantum geometrodynamics program. Our approach introduces a lattice regularization of the hypersurface deformation algebra, such that each lattice site carries a set of canonical variables given by the components of the spatial metric and the corresponding conjugate momenta. In order to quantize this theory, we describe a representation of the canonical commutation relations that enforces the positivity of the operators q_{ab} s^a s^b for all choices of s. Moreover, symmetry of q_{ab} and p^{ab} is ensured. This reflects the physical requirement that the spatial metric should be a positive definite, symmetric tensor. To achieve this end, we resort to the Cholesky decomposition of the spatial metric into upper triangular matrices with positive diagonal entries. Moreover, our Hilbert space also carries a representation of the vielbein fields and naturally separates the physical and gauge degrees of freedom. Finally, we introduce a generalization of the Weyl quantization for our representation. We want to emphasize that our proposed methodology is amenable to applications in other fields of physics, particularly
in scenarios where the configuration space is restricted by complicated relationships among the degrees of freedom.

Conformal Image Viewpoint Invariant

In this paper, we introduce an invariant by image viewpoint changes by applying an important theorem in conformal geometry stating that every surface of the Minkowski space R3,1 leads to an invariant by conformal transformations. For this, we identify the domain of an image to the disjoint union of horospheres ∐αHα of R3,1 by means of the powerful tools of the conformal Clifford algebras. We explain that every viewpoint change is given by a planar similarity and a perspective distortion encoded by the latitude angle of the camera. We model the perspective distortion by the point at infinity of the conformal model of the Euclidean plane described by D. Hestenesand we clarify the spinor representations of the similarities of the Euclidean plane. This leads us to represent the viewpoint changes by conformal transformations of ∐αHα for the Minkowski metric of the ambient space.

Green commutes: Assessing the associations between green space exposure along GPS-track commuting routes and adults’ self-perceived stress

BackgroundHabitual commutes through green space may be associated with health promotion, including stress alleviation. However, few studies have assessed green space exposure during commuting with stress levels, and none have tracked commuters’ actual routes. AimTo assess 1) the association between commuting through green space and people's self-perceived stress, 2) whether the association is moderated by the transport mode, and 3) whether the association depends on differing green space operationalizations and buffer sizes. MethodsThis cross-sectional study used questionnaires and global positioning system data from Dutch adults aged 18–65 (N = 275). The Perceived Stress Scale was used to measure people's stress levels. Green space was measured by calculating the green space percentage from land use data (GREENLU), and by using normalized difference vegetation index (NDVI) obtained from Sentinel-2 (NDVISE) and Landsat-8 (NDVILS) satellite imagery. Buffers of 50, 100, and 250 m along commuting routes were applied to assess subjects’ environmental exposures. Associations were estimated using ordinary least squares regression models. ResultsCovariate-adjusted regressions showed that GREENLU was significantly but positively associated with stress levels, regardless of the buffer size. By contrast, the NDVI measures consistently showed null associations. We observed a positive association between GREENLU and stress levels among active commuters within the 250 m buffer in stratified analyses; however, associations were null for passive commuters across all green space measures and buffer sizes. ConclusionsOur findings suggest a counterintuitive positive association between increased exposure to green space during daily commutes and people's stress levels. These associations possibly may depend on the selected green space metric, the buffer sizes, and the commuting mode considered. The behavioral aspects of how people experience green environments, including commuting, may contribute to their impact on health.

Missing Wedge Completion via Unsupervised Learning with Coordinate Networks.

Cryogenic electron tomography (cryoET) is a powerful tool in structural biology, enabling detailed 3D imaging of biological specimens at a resolution of nanometers. Despite its potential, cryoET faces challenges such as the missing wedge problem, which limits reconstruction quality due to incomplete data collection angles. Recently, supervised deep learning methods leveraging convolutional neural networks (CNNs) have considerably addressed this issue; however, their pretraining requirements render them susceptible to inaccuracies and artifacts, particularly when representative training data is scarce. To overcome these limitations, we introduce a proof-of-concept unsupervised learning approach using coordinate networks (CNs) that optimizes network weights directly against input projections. This eliminates the need for pretraining, reducing reconstruction runtime by 3-20× compared to supervised methods. Our in silico results show improved shape completion and reduction of missing wedge artifacts, assessed through several voxel-based image quality metrics in real space and a novel directional Fourier Shell Correlation (FSC) metric. Our study illuminates benefits and considerations of both supervised and unsupervised approaches, guiding the development of improved reconstruction strategies.

Moduli spaces of Ricci positive metrics in dimension five

Moduli spaces of Ricci positive metrics in dimension five

Evaluating high-resolution computed tomography derived 3-D joint space metrics of the metacarpophalangeal joints between rheumatoid arthritis and age- and sex-matched control participants.

Rheumatoid arthritis (RA) is commonly characterized by joint space narrowing. High-resolution peripheral quantitative computed tomography (HR-pQCT) provides unparalleled in vivo visualization and quantification of joint space in extremity joints commonly affected by RA, such as the 2nd and 3rd metacarpophalangeal joints. However, age, sex, and obesity can also influence joint space narrowing. Thus, this study aimed to determine whether HR-pQCT joint space metrics could distinguish between RA patients and controls, and determine the effects of age, sex and body mass index (BMI) on these joint space metrics. HR-pQCT joint space metrics (volume, width, standard deviation of width, maximum/minimum width, and asymmetry) were acquired from RA patients and age-and sex-matched healthy control participants 2nd and 3rd MCP joints. Joint health and functionality were assessed with ultrasound (i.e., effusion and inflammation), hand function tests, and questionnaires. HR-pQCT-derived 3D joint space metrics were not significantly different between RA and control groups (p > 0.05), despite significant differences in inflammation and joint function (p < 0.05). Joint space volume, mean joint space width (JSW), maximum JSW, minimum JSW were larger in males than females (p < 0.05), while maximum JSW decreased with age. No significant association between joint space metrics and BMI were found. HR-pQCT did not detect group level differences between RA and age-and sex-matched controls. Further research is necessary to determine whether this is due to a true lack of group level differences due to well-controlled RA, or the inability of HR-pQCT to detect a difference.

Moduli space of bi-invariant metrics

In this work, we focus on describing the space of bi-invariant metrics in a Lie group up to isometry. i.e, metrics invariant under both left and right translations. We show that BI\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {BI}$$\\end{document}, the moduli space of bi-invariant metrics, is an orbifold. Moreover, we give an explicit description of this orbifold, and of EBI\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {EBI}$$\\end{document}, the space of bi-invariant metrics equivalent under isometries and scalar multiplies.

Thurston’s asymmetric metric on the space of singular flat metrics with a fixed quadrangulation

Consider a compact surface equipped with a fixed quadrangulation. One may identify each quadrangle on the surface with a Euclidean rectangle to obtain a singular flat metric on the surface with conical singularities. We call such a singular flat metric a rectangular structure. We study a metric on the space of unit area rectangular structures which is analogous to Thurston’s asymmetric metric on the Teichmüller space of a surface of finite type. We prove that the distance between two rectangular structures is equal to the logarithm of the maximum of ratios of edges of these rectangular structures. We give a sufficient condition for a path between two points of the this Teichmüller space to be geodesic and we prove that any two points of this space can be joined by a geodesic. We also prove that this metric is Finsler and give a formula for the infinitesimal weak norm on the tangent space of each point. We identify the space of unit area rectangular structures with a submanifold of a Euclidean space and we show that the subspace topology and the topology induced by the metric we introduced coincide. We show that the space of unit area rectangular structures on a surface with a fixed quadrangulation is in general not complete.

The Liouville current of holomorphic quadratic differential metrics

In this paper we study the Liouville current of flat cone metrics coming from a holomorphic quadratic differential. Anja Bankovic and Christopher J. Leininger proved in Bankovic and Leininger (Trans Am Math Soc 370:1867–1884, 2018) that for a fixed closed surface, there is an injection map from the space of flat cone metrics to the space of geodesic currents. We manage to show that metrics coming from holomorphic quadratic differentials can be distinguished from other flat metrics by just looking at the geodesic currents. The key idea is to analyze the support of Liouville current, which is a topological invariant independent of the metric, and get information about cone angles and holonomy. The holonomy part involves some subtlety of relationship between singular foliation and geodesic lamination. We also obtain a new proof of a classical result that almost all simple geodesics of a quadratic differential metric will be dense in the surface. Furthermore, for other flat cone metrics, there is no simple dense geodesic.

The impact of ozone on Earth-like exoplanet climate dynamics: the case of Proxima Centauri b

ABSTRACT The emergence of the JWST and the development of other advanced observatories (e.g. ELTs, LIFE, and HWO) marks a pivotal moment in the quest to characterize the atmospheres of Earth-like exoplanets. Motivated by these advancements, we conduct theoretical explorations of exoplanetary atmospheres, focusing on refining our understanding of planetary climate and habitability. Our study investigates the impact of ozone on the atmosphere of Proxima Centauri b in a synchronous orbit, utilizing coupled climate chemistry model simulations and dynamical systems theory. The latter quantifies compound dynamical metrics in phase space through the inverse of co-persistence (θ) and co-dimension (d), of which low values correspond to stable atmospheric states. Initially, we scrutinized the influence of ozone on temperature and wind speed. Including interactive ozone [i.e. coupled atmospheric (photo)chemistry] reduces the hemispheric difference in temperature from 68 °K to 64 °K, increases (∼+7 °K) atmospheric temperature at an altitude range of ∼20–50 km, and increases variability in the compound dynamics of temperature and wind speed. Moreover, with interactive ozone, wind speed during highly temporally stable states is weaker than for unstable ones, and ozone transport to the nightside gyres during unstable states is enhanced compared to stable ones (∼+800 DU). We conclude that including interactive ozone significantly influences Earth-like exoplanets' chemistry and climate dynamics. This study establishes a novel pathway for comprehending the influence of photochemical species on the climate dynamics of potentially habitable Earth-like exoplanets. We envisage an extension of this framework to other exoplanets.

Assessing life expectancy disparities in Chicago with a deep dive into green space

BackgroundFew studies have examined the association between green space and life expectancy, and only one study has looked at that data for small areas in the United States. These studies have found a positive association between green space and life expectancy. This study investigates the association between green space and small-area life expectancy in Chicago, a city in the United States with distinct environmental and socioeconomic characteristics and the largest U.S. life expectancy gap between neighborhoods. MethodsWe conducted a spatial analysis using the green space metrics of normalized difference vegetation index (NDVI), park access, and tree canopy. Additionally, we included 18 lifestyle, environmental, and social covariates. Diagnostic analytics included multiple linear regression, Moran I test for spatial dependence, and principal component analysis with CAR Gaussian model. We performed a simulation to predict life expectancy gain if NDVI was brought to median levels. ResultsOur study revealed that after adjusting for covariates and spatial dependence (Moran I test p = 0.012) with a CAR Gaussian model, NDVI (credible interval = 0.292-0.894) demonstrated significance at a 5 % level, while tree canopy (credible interval = -0.404-0.109) and park access (credible interval = -0.258-0.161) did not. Excluding non-significant variables (tree canopy and park access) from the CAR Gaussian model, the median effect of NDVI on life expectancy was 0.491 (95 % credible interval: 0.24–0.718). Projecting this effect to tracts below the median NDVI predicted a gain of 476,000 years of life expectancy for Chicago residents. ConclusionThis study provides evidence that green space and LE are positively correlated, suggesting the potential to increase green space to address current LE gaps. Future research should investigate the mechanisms by which green space correlates to lifespan.

Navigating the Manifold of Translucent Appearance

AbstractWe present a perceptually‐motivated manifold for translucent appearance, designed for intuitive editing of translucent materials by navigating through the manifold. Classic tools for editing translucent appearance, based on the use of sliders to tune a number of parameters, are challenging for non‐expert users: These parameters have a highly non‐linear effect on appearance, and exhibit complex interplay and similarity relations between them. Instead, we pose editing as a navigation task in a low‐dimensional space of appearances, which abstracts the user from the underlying optical parameters. To achieve this, we build a low‐dimensional continuous manifold of translucent appearance that correlates with how humans perceive this type of materials. We first analyze the correlation of different distance metrics in image space with human perception. We select the best‐performing metric to build a low‐dimensional manifold, which can be used to navigate the space of translucent appearance. To evaluate the validity of our proposed manifold within its intended application scenario, we build an editing interface that leverages the manifold, and relies on image navigation plus a fine‐tuning step to edit appearance. We compare our intuitive interface to a traditional, slider‐based one in a user study, demonstrating its effectiveness and superior performance when editing translucent objects.

[formula omitted] gravity theory in Einstein space background and causality violation

In this paper, we exploration a Petrov type-N vacuum solution to Einstein’s field equations, while incorporating a negative cosmological constant (Λ<0) within the framework of modified gravity theories. This solution intriguingly accommodates closed time-like curves at a particular moment in time, effectively violates the causality condition, thus acts as a time-machine model. A key observation is that the determinant of the Ricci tensor Rμν for this particular Einstein space metric differs from zero. This noteworthy finding suggests to the existence of an anti-curvature tensor defined Aμν=Rμν−1 and hence, an anti-curvature scalar A=gμνAμν, which is introduced with the Lagrangian of the system, thereby giving rise to as Ricci-inverse gravity theory. We consider class-I models of Ricci-inverse gravity, where the function f=f(R,A)=(R+κA) with κ is the coupling constant. We demonstrate that this Einstein space metric serves as a vacuum solution with a negative modified cosmological constant within the framework of Ricci-inverse gravity. Consequently, the violation of causality persists within this new gravity theory as well. Moreover, we solve the modified field equations by considering matter content other than vacuum and demonstrate that the energy-density and isotropic pressure satisfies the equation ρ=−p=3κΛ.

Reference Data for Diagnosis of Spondylolisthesis and Disc Space Narrowing Based on NHANES-II X-rays.

Robust reference data, representing a large and diverse population, are needed to objectively classify measurements of spondylolisthesis and disc space narrowing as normal or abnormal. The reference data should be open access to drive standardization across technology developers. The large collection of radiographs from the 2nd National Health and Nutrition Examination Survey was used to establish reference data. A pipeline of neural networks and coded logic was used to place landmarks on the corners of all vertebrae, and these landmarks were used to calculate multiple disc space metrics. Descriptive statistics for nine SPO and disc metrics were tabulated and used to identify normal discs, and data for only the normal discs were used to arrive at reference data. A spondylolisthesis index was developed that accounts for important variables. These reference data facilitate simplified and standardized reporting of multiple intervertebral disc metrics.

Correction: Intersection theory and volumes of moduli spaces of flat metrics on the sphere

Correction: Intersection theory and volumes of moduli spaces of flat metrics on the sphere

Poincaré Differential Privacy for Hierarchy-Aware Graph Embedding

Hierarchy is an important and commonly observed topological property in real-world graphs that indicate the relationships between supervisors and subordinates or the organizational behavior of human groups. As hierarchy is introduced as a new inductive bias into the Graph Neural Networks (GNNs) in various tasks, it implies latent topological relations for attackers to improve their inference attack performance, leading to serious privacy leakage issues. In addition, existing privacy-preserving frameworks suffer from reduced protection ability in hierarchical propagation due to the deficiency of adaptive upper-bound estimation of the hierarchical perturbation boundary. It is of great urgency to effectively leverage the hierarchical property of data while satisfying privacy guarantees. To solve the problem, we propose the Poincar\'e Differential Privacy framework, named PoinDP, to protect the hierarchy-aware graph embedding based on hyperbolic geometry. Specifically, PoinDP first learns the hierarchy weights for each entity based on the Poincar\'e model in hyperbolic space. Then, the Personalized Hierarchy-aware Sensitivity is designed to measure the sensitivity of the hierarchical structure and adaptively allocate the privacy protection strength. Besides, Hyperbolic Gaussian Mechanism (HGM) is proposed to extend the Gaussian mechanism in Euclidean space to hyperbolic space to realize random perturbations that satisfy differential privacy under the hyperbolic space metric. Extensive experiment results on five real-world datasets demonstrate the proposed PoinDP’s advantages of effective privacy protection while maintaining good performance on the node classification task.

A Note about Isotopy and Concordance of Positive Scalar Curvature Metrics on Compact Manifolds with Boundary

We study notions of isotopy and concordance for Riemannian metrics on manifolds with boundary and, in particular, we introduce two variants of the concept of minimal concordance, the weaker one naturally arising when considering certain spaces of metrics defined by a suitable spectral ''stability'' condition. We develop some basic tools and obtain a rather complete picture in the case of surfaces.