Gaussian processes adapted for Riemannian manifolds using gauge-independent kernels.
problem Deploying Gaussian processes on non-Euclidean domains like Riemannian manifolds.
method Developed techniques to generalize Gaussian processes to vector fields on Riemannian manifolds using gauge-independent kernels.
result Enabled training of vector-valued Gaussian processes on Riemannian manifolds using standard Gaussian process methods.
We extend Gaussian Differential Privacy to curved Riemannian manifolds.
problem Extending Gaussian Differential Privacy to curved spaces.
method Developed a Riemannian Gaussian distribution using the Bishop-Gromov theorem and a MCMC-based algorithm.
result Achieved Gaussian Differential Privacy on general Riemannian manifolds with bounded Ricci curvature.
New Riemannian geometry for Compound Gaussian distributions applied to efficient change detection.
problem Change detection in multivariate image times series.
method Developed a recursive approach based on Riemannian optimization.
result Optimal performance achieved with computational efficiency.
New Gaussian processes for Riemannian manifolds enable uncertainty quantification.
problem Modeling functions on Riemannian manifolds with uncertainty.
method Generalized Matérn Gaussian processes on compact manifolds via spectral theory.
result Efficient training of Riemannian Matérn Gaussian processes using scalable techniques.
Gaussian kernels on complex manifolds are never positive definite.
problem Analyzing positive definiteness of Gaussian kernels on non-simply-connected Riemannian manifolds.
method Combining recent preprint analysis and classical Riemannian geometry comparison theorems.
result Gaussian kernels are never positive definite on non-simply-connected closed Riemannian manifolds.
This study describes the Fisher-Rao metric on Gaussian measures in infinite-dimensional spaces.
problem Understanding the Fisher-Rao metric in infinite-dimensional Gaussian settings.
method Explicit description and generalization of finite-dimensional quantities to infinite-dimensional Hilbert spaces.
result The Fisher-Rao metric and related geometric quantities generalize from finite to infinite dimensions.
The paper introduces a differentially private method for optimization on Riemannian manifolds.
problem Differential privacy in optimization constrained to Riemannian manifolds.
method Adding Gaussian noise to the Riemannian gradient on the tangent space, with privacy and utility guarantees.
result Privacy and utility guarantees for differentially private Riemannian optimization.
Sharp sub-Gaussian bounds for subsolutions of Trudinger's equation on Riemannian manifolds.
problem Bounding weak subsolutions of Trudinger's equation on Riemannian manifolds.
method Proving sub-Gaussian upper bounds for weak subsolutions.
result The upper bounds are sharp for specific classes of manifolds, including \(\mathbb{R}^{n}\).
The study finds complete translating solitons for certain powers of Gaussian curvature in Riemannian products.
problem Exploring translating solitons in Riemannian products with powers of Gaussian curvature.
method Investigating K α K^α K α -flows in Riemannian products M i m e s R M imes\mathbb R M im es R for M = R n , S n , H F m M=\mathbb R^n, \mathbb S^n, \mathbb{H}_{\mathbb F}^m M = R n , S n , H F m . result Existence of complete rotational translating solitons for certain values of α α α in M i m e s R M imes\mathbb R M im es R . Study curvature and torsion in Gaussian distribution's dual coordinate system.
problem Characterize geometric invariants of Gaussian distribution.
method Investigate Riemannian curvature and torsion in a dual coordinate system of Gaussian distribution.
result Explicitly give Amari formulas in the new coordinate system.
New algorithm learns HMM parameters on Riemannian manifolds.
problem Learning hidden Markov models on non-Euclidean spaces.
method Geometric method of moments algorithm for Riemannian manifolds.
result Significantly improved speed and accuracy compared to existing methods.
Bayesian neural networks approximate Gaussian, this method adapts to non-Gaussian posteriors.
problem Bayesian neural networks struggle with non-Gaussian posteriors, leading to poor performance.
method Proposes a Riemannian Laplace approximation to adapt to the shape of the true posterior.
result Consistently improves over conventional Laplace approximation across tasks.
The paper studies the asymptotic expansion of Gaussian integral operators on Riemannian submanifolds.
problem Analyzing the asymptotic behavior of Gaussian integral operators on Riemannian submanifolds.
method Deriving a full asymptotic expansion of the Gaussian integral operator and computing the first-order correction term.
result Explicit computation of the first-order correction term in terms of mean curvature vector and scalar curvature.
This study examines Gaussian processes on Riemannian manifolds and proves contraction rates.
problem Comparing intrinsic vs. extrinsic Gaussian processes on Riemannian manifolds.
method Proves optimal contraction rates for intrinsic Matérn Gaussian processes on compact Riemannian manifolds.
result Intrinsic Gaussian processes on Riemannian manifolds achieve better performance than extrinsic ones.
New method improves Gaussian Mixture Model fitting speed.
problem Slow convergence of EM algorithm for complex data.
method Riemannian Newton Trust-Region method for Gaussian Mixture Models.
result Outperforms existing methods in runtime and iterations.
Study on estimating distances between covariance operators and Gaussian processes.
problem Estimating distances between covariance operators and Gaussian processes.
method Riemannian distances, concentration results for Hilbert space-valued random variables, RKHS covariance and cross-covariance operators.
result Both distances converge in the Hilbert-Schmidt norm and can be consistently and efficiently estimated.
Paper estimates Gaussian curvature of minimal graphs in a specific manifold.
problem Estimating Gaussian curvature of minimal graphs in M i m e s R M imes\mathbb{R} M im es R . method Using Weierstrass representation via ℘ − \wp- ℘ − harmonic mappings and Schwarz lemma type results. result Proves Schwarz lemma type and Heinz type results for harmonic mappings.
Gaussian processes adapted for non-Euclidean spaces enhance decision-making.
problem Applying Gaussian processes in non-Euclidean spaces.
method Developed pathwise conditioning and Gaussian process models over non-Euclidean spaces.
result Efficient Gaussian process models for non-Euclidean spaces.
In this paper we consider the Gaussian thermostat ray transform on both closed Riemannian surfaces and compact Riemannian surfaces with boundary. We establish certain results on the injectivity of the thermostat ray transform and the surjectivity of its adjoint.
Geometric Gaussian approximations capture any distribution.
problem Approximating complex probability distributions.
method Geometric Gaussian approximations through diffeomorphisms or exponential maps.
result Geometric Gaussian approximations are universal, capturing any distribution.
Proofs Fisher-Rao distance on Gaussian covariance manifold.
problem Proving Fisher-Rao distance on Gaussian covariance manifold.
method Basic Riemannian geometry.
result Proof of Fisher-Rao distance on covariance cone.
Calculates spinor heat flow using Gaussian-Grassmann integrals.
problem Computing the spinor heat flow symbol.
method Getzler calculus and Gaussian-Grassmann integrals.
result Computed the spinor heat flow symbol.
Proposes a Riemannian optimization for policy improvement in MDPs.
problem Optimizing policy functions in Markov decision processes (MDPs).
method Riemannian proximal optimization algorithm with Gaussian mixture model (GMM).
result Guaranteed convergence and efficacy demonstrated in preliminary experiments.
We introduce Gaussian-type measures on the manifold of all metrics with a fixed volume form on a compact Riemannian manifold of dimension ≥ 3 \geq 3 ≥ 3 . For this random model we compute the characteristic function for the L 2 L^2 L 2 (Ebin) distance to the reference metric. In the Appendix, we study Lipschitz-type distance betwee…
The space of Gaussian measures on a Euclidean space is geodesically convex in the L 2 L^2 L 2 -Wasserstein space. This space is a finite dimensional manifold since Gaussian measures are parameterized by means and covariance matrices. By restricting to the space of Gaussian measures inside the L 2 L^2 L 2 -Wasserstein space, we manag…
Random Gaussian fields on 4D Riemannian manifolds with conformal invariance.
problem Characterizing and analyzing Gaussian fields on 4D Riemannian manifolds.
method Constructing and analyzing co-biharmonic Gaussian fields with covariance kernels defined by the Paneitz operator.
result Rigorous derivation of quantum Liouville measure for ∣ γ ∣ < 8 |γ|<\sqrt8 ∣ γ ∣ < 8 . New test detects sparse alternatives in Gaussian random fields.
problem Detecting sparse alternatives in Gaussian random fields.
method Ad-hoc Kac Rice formula for second maximum distribution, exact spacing test.
result Exact t t t -spacing test for high power in detecting sparse alternatives. A new prior for VAEs improves model capacity by allowing a more flexible latent space.
problem Standard Gaussian priors in VAEs limit model capacity and performance.
method Proposed a Riemannian Brownian motion prior over a Riemannian structure of the latent space.
result The new prior significantly increases model capacity with only one additional scalar parameter.
Paper solves a key problem in learning from high-dimensional covariance matrices.
problem Computing normalizing factors for Riemannian Gaussian distributions on high-dimensional covariance matrices.
method Equivalence with random matrix theory and log-normal matrix ensembles to approximate normalizing factors.
result Efficient approximation of normalizing factors with decreasing error as dimension increases.
Let M be a compact Riemannian manifold and E a Riemannian vector bundle on M. We look for hypersurfaces of E with a prescribed vertical Gaussian curvature. In trying to solve this problem fibre-wise, we loose the regularity of the resulting solution. To unsure the smoothness of the solution, we construct it as a radial…
Synthetic scalar curvature defined via Gaussian integrals, applied to manifolds and flows.
problem Defining scalar curvature for non-smooth spaces and flows.
method Gaussian integral approach to scalar curvature, applied to manifolds and flows.
result Characterizes Ricci flows as minimal super-Ricci flows.
The paper studies heat kernels on weighted Riemannian manifolds with curvature bounds.
problem Estimating heat kernels on weighted Riemannian manifolds with lower Ricci curvature bounds.
method Establishing parabolic Harnack inequalities, proving Gaussian bounds for heat kernels, and constructing Li-Yau-type gradient estimates.
result Gaussian upper and lower bounds for the heat kernel, Liouville theorem, uniqueness property, and eigenvalue bounds.
New method estimates Riemannian derivatives from noisy function evaluations.
problem Optimizing functions on Riemannian manifolds with noisy data.
method Riemannian Gaussian smoothing for gradient and Hessian estimation.
result Oracle complexity independent of ambient dimension.
Proposes a neural network for recognizing 3D skeleton-based interactions.
problem Recognizing two-person interactions from 3D skeleton sequences.
method Uses Gaussian distributions and Riemannian geometry of SPD matrices and matrix groups.
result Achieves competitive results on three benchmarks for 3D human activity understanding.
The paper presents a probabilistic framework for SPD matrices in machine learning.
problem Machine learning on SPD matrices is fragmented; this paper aims to unify it.
method Unified probabilistic framework using Gaussian distributions and Bayes classifiers.
result Different SPD machine learning tools can be reinterpreted and extended using Gaussian distributions.
Convex surfaces derived from specific Riemannian manifolds with high regularity.
problem Proving convexity of surfaces derived from Riemannian manifolds.
method Analyzing solutions to the very weak Monge-Ampère equation.
result Proved convexity of weakly regular surfaces with nonnegative intrinsic curvature.
The paper connects Riemannian Gaussian distributions to random matrix theory and diffusion kernels.
problem Analyzing Riemannian Gaussian distributions on symmetric spaces.
method Analytical computation of marginals using orthogonal and skew orthogonal polynomials, and diffusion kernels.
result Riemannian Gaussian distributions are random matrix types, and their probability density functions can be computed analytically.
This technical report presents pseudo-code for a Riemannian manifold Hamiltonian Monte Carlo (RMHMC) method to efficiently simulate samples from N N N -dimensional posterior distributions p ( x ∣ y ) p(x|y) p ( x ∣ y ) , where x ∈ R N x \in R^N x ∈ R N is drawn from a Gaussian Process (GP) prior, and observations y n y_n y n are independent given x n x_n x n . Sufficient…
The study classifies parallel mean curvature spheres in a sphere-hyperbolic product space.
problem Understanding surfaces with parallel mean curvature in a specific Riemannian product space.
method Analyzing the holomorphic quadratic differential and topological constraints.
result Classification of all parallel mean curvature spheres with vanishing differential.
New geometric SDEs and discretizations on Riemannian manifolds with error bounds.
problem Modeling diffusion processes on Riemannian manifolds with geometric SDEs.
method Introduced a new construction of geometric SDEs and provided non-asymptotic error bounds.
result First non-asymptotic error bound for geometric Euler-Murayama discretization.
The Gaussian kernel is never positive-definite on Riemannian symmetric spaces.
problem Proving the non-positive-definiteness of the Gaussian kernel on non-Euclidean symmetric spaces.
method Developed new geometric and analytical arguments to rigorously characterize the positive-definiteness of the Gaussian kernel.
result L p ^{\!\scriptscriptstyle p} p - \hspace{0.02cm} Godement theorems provide necessary and sufficient conditions for positive-definiteness. Authors calculate limits of curvatures on surfaces in sub-Riemannian manifolds.
problem Calculating limits of Gaussian and normal curvatures on surfaces in sub-Riemannian manifolds.
method Utilized Riemannian approximations scheme in Heisenberg group to calculate limits of curvatures.
result Obtained Gauss-Bonnet theorem as a limit of theorems in approximations schemes.
The paper provides precise estimates for isoperimetric inequalities on weighted manifolds.
problem Quantitative isoperimetric inequalities on weighted Riemannian manifolds.
method Analyzes L 1 L^1 L 1 , L p L^p L p , and W 2 W_2 W 2 estimates for the push-forward of measures. result Close approximation of the guiding function's push-forward to Gaussian measure.
We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampere type. These two problems are: the local isometric embedding problem for two-dimensional Riemannian manifolds, and the problem of locally prescribed Gaussian curvature for surfaces in …
We consider maximum likelihood estimation for Gaussian Mixture Models (Gmms). This task is almost invariably solved (in theory and practice) via the Expectation Maximization (EM) algorithm. EM owes its success to various factors, of which is its ability to fulfill positive definiteness constraints in closed form is of …
The study proves the finiteness of moments for Gaussian field zeros and critical points.
problem Finiteness of moments for Gaussian field zeros and critical points.
method Definition and study of multijets, construction of p-multijet bundles.
result Linear statistics of Gaussian field zeros have finite p-th moments for p ≥ 1.
We consider small-time asymptotics for diffusion processes conditioned by their initial and final positions, under the assumption that the diffusivity has a sub-Riemannian structure, not necessarily of constant rank. We show that, if the endpoints are joined by a unique path of minimal energy, and lie outside the sub-R…
The paper estimates eigenvalues for specific differential operators on curved spaces.
problem Estimating eigenvalues for a class of elliptic differential operators on Riemannian manifolds.
method Analyzes eigenvalue estimates for a broader class of elliptic differential operators in divergence form.
result Provides eigenvalue estimates for Gaussian shrinking solitons and specific domains.