We review the well-known slice theorem of Ebin for the action of the diffeomorphism group on the space of Riemannian metrics of a closed manifold. We present advances in the study of the spaces of Riemannian metrics, and produce a more concise proof for the existence of slices.
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48 results for “slice metric”
Study of Riemannian geometry on quaternionic unit ball linked to Sp(1,1) group.
problem Understanding the geometry induced by slice Riemannian metric.
method Developed Lie theoretic study, computed isometry group, compared with quaternionic Poincaré geometry.
result Isometry group of slice Riemannian metric is built from symmetries of Sp(1,1) group.
Let X be a G-space such that the orbit space X/G is metrizable. Suppose a family of slices is given at each point of X. We study a construction which associates, under some conditions on the family of slices, with any metric on X/G an invariant metric on X. We show also that a family of slices with the required propert…
The paper explores statistical and topological properties of sliced probability divergences.
problem Understanding the topological, statistical, and computational consequences of slicing divergences.
method Deriving theoretical properties of sliced probability divergences, including metric axioms preservation and weak continuity.
result Sliced divergences share similar topological properties and have stable sample complexity.
Proposes a new distance metric for multi-marginal optimal transport.
problem Computational scalability in multi-marginal optimal transport.
method Random one-dimensional projections to construct sliced multi-marginal Wasserstein distance.
result Sliced multi-marginal Wasserstein distance is a metric with dimension-free sample complexity.
In this paper, we proved a rigidity theorem of the Hodge metric for concave horizontal slices and a local rigidity theorem for the monodromy representation.
Improves point-cloud reconstruction by optimizing projections with self-attention.
problem Inefficient and non-metric projection methods for sliced Wasserstein distances.
method Proposes distributional sliced Wasserstein distance with self-attention for permutation-invariant and metric optimization.
result Self-attention amortized distributional projection optimization achieves better performance in point-cloud reconstruction.
We give a topological condition for a generic sliced space to be globally hyperbolic, without any hypothesis on the lapse function, shift function and spatial metric.
We establish a general slice theorem for the action of a locally convex Lie group on a locally convex manifold, which generalizes the classical slice theorem of Palais to infinite dimensions. We discuss two important settings under which the assumptions of this theorem are fulfilled. First, using Glöckner's inverse fun…
The paper decomposes metrics on manifolds with boundaries.
problem Characterizing Riemannian metrics on manifolds with boundaries.
method Koiso-type decomposition and Ebin-type slice theorems.
result Characterization of relative Einstein metrics.
The paper studies CMC foliations and their conformal aspects on Riemannian manifolds.
problem Understanding and normalizing CMC foliations on conformally compact manifolds.
method Non-linear PDEs and conformal transformations.
result Locally, any slicing can be made into a CMC foliation by conformal changes.
Introduces MSW distances to improve SW metrics.
problem Redundant projections in SW distance.
method Imposes Markov structure on projecting directions.
result MSW distances improve SW metrics.
Gromov-Hausdorff convergence of time-slices of singular Ricci flows
problem Gromov-Hausdorff convergence of time-slices of singular Ricci flows
method Completion of singular Ricci flow with respect to a natural spacetime distance
result Gromov-Hausdorff convergence at the first singular time
New slicing methods speed up Gaussian mixture Wasserstein distance computations.
problem High computational cost of the mixture Wasserstein distance.
method Slicing-based approximations to reduce computational complexity.
result Significant reduction in computational complexity while preserving key properties.
Optimal transport (\OT) theory defines a powerful set of tools to compare probability distributions. \OT~suffers however from a few drawbacks, computational and statistical, which have encouraged the proposal of several regularized variants of OT in the recent literature, one of the most notable being the \textit{slice…
A new metric for comparing measures on tree systems reduces computational burden.
problem Heavy computation in Optimal Transport problems.
method Introducing tree systems and a novel metric (Tree-Sliced Wasserstein distance on Systems of Lines, TSW-SL).
result TSW-SL performs favorably compared to Sliced Wasserstein and its variants.
Proposes an energy-based sliced Wasserstein distance for improved probability measure comparison.
problem Inefficiencies and limitations in existing sliced Wasserstein distance approaches.
method Introduces an energy-based slicing distribution for better performance and stability.
result Demonstrates superior performance of the EBSW distance in various applications.
A new metric HSW derived from hierarchical Radon Transform addresses computational bottlenecks in sliced Wasserstein.
problem Computational inefficiency of sliced Wasserstein in high-dimensional settings with few supports.
method Hierarchical Radon Transform (HRT) and bottleneck projections to reduce projection number.
result HSW metric derived from HRT is computationally efficient and maintains metric properties.
In real-world machine learning applications, data subsets correspond to especially critical outcomes: vulnerable cyclist detections are safety-critical in an autonomous driving task, and "question" sentences might be important to a dialogue agent's language understanding for product purposes. While machine learning mod…
This work investigates the properties of Gaussian-smoothed sliced divergences for comparing distributions.
problem Comparing probability distributions while preserving privacy.
method Investigates the theoretical properties of Gaussian-smoothed sliced Wasserstein distance and generalized versions.
result Gaussian smoothed sliced Wasserstein distance converges with a rate of \(O(n^{-1/2})\).
A new approach simplifies Sliced-Wasserstein distances to improve learning performance.
problem The concentration of measure phenomenon makes random projections uninformative in high dimensions.
method Propose rescaling the 1D Wasserstein distance to make all slices equally informative.
result The classical Sliced-Wasserstein, properly configured, can match or surpass complex variants.
A new method for comparing image probability measures using convolution operators.
problem Efficiently comparing images using conventional sliced Wasserstein methods.
method Proposed convolution sliced Wasserstein (CSW) methods with stride, dilation, and non-linear activation.
result CSW demonstrates favorable performance over conventional sliced Wasserstein in image comparison and deep generative modeling.
New method for reducing dimensions of distributional data.
problem Nonlinear sufficient dimension reduction for distribution-on-distribution regression.
method Building universal kernels on metric spaces to characterize conditional independence.
result Method outperforms competing methods in synthetic and real data applications.
Develops slicing method to prove rigidity of scalar curvature on manifolds with boundary.
problem Understanding positive scalar curvature metrics on manifolds with boundary.
method Minimal slicing via capillary hypersurfaces to prove rigidity statements.
result Proves rigidity statement in dimension 4 for specific geometric conditions.
Hierarchical Hybrid Sliced Wasserstein: A Scalable Metric for Heterogeneous Joint Distributionscs.CV
Extends SW and GSW to compare heterogeneous joint distributions.
problem Limited applicability of SW and GSW to heterogeneous joint distributions.
method Introduces HHRT and PGRT to extend SW and GSW.
result H2SW distance for heterogeneous joint distributions.
A new variational inference method using sliced Wasserstein distance is proposed.
problem The inefficiency and unreasonable properties of Kullback-Leibler divergence.
method Minimizing sliced Wasserstein distance, a valid metric from optimal transport.
result The proposed method approximates the unnormalized distribution efficiently and without requiring a tractable density function.
A new Wasserstein distance method for comparing incomparable distributions.
problem Comparing distributions that are not supported on the same metric space.
method Distributional slicing, embeddings, and closed-form computation of Wasserstein distance.
result HWD preserves properties like rotation-invariance and can be efficiently learned.
Max-sliced Wasserstein metric reduces high-dimensional data to 1D for better estimation.
problem Curse of dimensionality in optimal transport.
method Introduces max-sliced Wasserstein metric to reduce high-dimensional problems to 1D.
result Uniform ratio bounds of empirical measures on RKHS concentrate uniformly fast at parametric rates.
We show that the regular Slodowy slice to the sum of two semisimple adjoint orbits of is isomorphic to the deformation of the -singularity if , the Dancer deformation of the double cover of the Atiyah-Hitchin manifold if , and to the Atiyah-Hitchin manifold itself if . For higher , such…
The Kasner metrics are among the simplest solutions of the vacuum Einstein equations, and we use them here to examine the conformal method of finding solutions of the Einstein constraint equations. After describing the conformal method's construction of constant mean curvature (CMC) slices of Kasner spacetimes, we turn…
New curvature concept shows certain manifolds can't have positive curvature.
problem Understanding manifolds that can't have positive curvature metrics.
method Introducing -intermediate curvature and using stable weighted slicings.
result Manifolds do not admit positive -intermediate curvature for .
New cosmological models with changing curvature slices.
problem Cosmological models with varying and sign-changing curvature.
method Constructing globally hyperbolic spacetimes with slices of constant curvature that can change sign.
result Shows at least one comoving observer disappearing in finite time.
Cohomogeneity one RCD-spacesmath.MG
Study cohomogeneity one RCD-spaces, proving structural results and constructing new examples.
problem Characterize and construct RCD-spaces with cohomogeneity one actions.
method Slice Theorem, construction from group diagrams, topological structural results.
result Classification of cohomogeneity one, non-collapsed RCD-spaces of essential dimension at most 4.
Geometric structures on quaternionic unit ball for slice regular Möbius transformations.
problem No new problem introduced.
method Introducing Hermitian, Riemannian, and Kähler-like structures on quaternionic unit ball using regular Möbius transformations.
result Geometric structures are natural generalizations of complex setup and solve problems not achieved by other geometries.
Generative adversarial nets (GANs) and variational auto-encoders have significantly improved our distribution modeling capabilities, showing promise for dataset augmentation, image-to-image translation and feature learning. However, to model high-dimensional distributions, sequential training and stacked architectures …
Proves equality in Minkowski inequality for static, flat manifolds.
problem Proving equality in Minkowski inequality for static, flat manifolds.
method Analyzes quasi-spherical metrics and static manifolds.
result Equality in Minkowski inequality achieved only by Schwarzschild space slices.
A new method for fast optimal transport using sliced Wasserstein generalized geodesics.
problem Computing optimal transport distances efficiently and accurately.
method Proposes a new proxy of squared Wasserstein distance based on one-dimensional projections.
result min-SWGG is an upper bound of Wasserstein distance with similar computational complexity.
Study shows inflation in 3+1D cosmologies with bounded scalar potential and specific symmetry.
problem Understanding inflation in 3+1D cosmologies with specific constraints.
method Mean curvature flow and asymptotic analysis of metric variations, stress-energy tensor, and inflaton field dynamics.
result Inflation occurs in 3+1D cosmologies with specific constraints, demonstrating it is possible with inhomogeneous initial conditions.
New method for summarizing Bayesian mixture models using sliced Wasserstein distances.
problem Estimating the mixing measure in nonparametric Bayesian mixture models.
method Decision-theoretic approach using sliced Wasserstein distances for Gaussian mixtures.
result Effective estimation of the mixing measure and mixture density.
A new method for distribution regression using sliced Wasserstein distance.
problem Learning functions over spaces of probabilities.
method Proposes an OT-based estimator using the Sliced Wasserstein distance.
result Proves universal consistency and excess risk bounds for the proposed estimator.
New KQEs improve probability metrics without mean function constraints.
problem Improving probability metrics without relying on mean function representations.
method Kernel quantile embeddings (KQEs) to construct new distances.
result KQEs offer a competitive alternative to MMD with near-linear cost.
New method improves Wasserstein distance for large-scale data.
problem High computational cost of Wasserstein distance for large-scale machine learning.
method Augmented Sliced Wasserstein Distances (ASWDs) using neural network mappings.
result ASWDs significantly outperform other Wasserstein variants in synthetic and real-world problems.
Injective X-ray transform on Heisenberg group for regular functions.
problem Injectivity of X-ray transform on sub-Riemannian manifolds.
method Group Fourier Transform and analysis of taming metrics.
result Sufficiently regular functions on Heisenberg group are determined by their line integrals.
We study rigidity of minimal two-spheres that locally maximize the Hawking mass on a Riemannian three-manifold with a positive lower bound on its scalar curvature. After assuming strict stability of , we prove that a neighborhood of it in is isometric to one of the deSitter-Schwarzschild metrics on $(- ε,ε)\…
Probability metrics have become an indispensable part of modern statistics and machine learning, and they play a quintessential role in various applications, including statistical hypothesis testing and generative modeling. However, in a practical setting, the convergence behavior of the algorithms built upon these dis…
On Shake Slice Knotsmath.GT
Study on shake slice knots and proves 0-shake slice knots are slice.
problem Understanding and characterizing shake slice knots.
method Exploring the relation to corks and proving slice properties.
result Proves 0-shake slice knots are slice.
-Shake Slice Implies Slicemath.GT
Proves certain knots are slice without shaking.
problem Identifying slice knots without using traditional methods.
method Direct proof for shake slice knots.
result Proves shake slice knots are slice.
We analyze critical points of the Sliced Wasserstein Distance for optimization stability.
problem Understanding the behavior of optimization algorithms for models trained with the Sliced Wasserstein Distance.
method Explicit perturbations and critical point analysis of the SW objective.
result Stable critical points of SW cannot concentrate on segments, providing optimization stability.