We classify translation surfaces in isotropic geometry with arbitrary constant isotropic Gaussian and mean curvature under the condition that at least one of translating curves lies in a plane.
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48 results for “Isotropic Gaussian”
Paper proves a theorem about constant mean curvature surfaces in isotropic 3-space.
problem Understanding constant mean curvature surfaces in isotropic 3-space.
method Value distribution theorem of Gaussian curvature applied to CMC surfaces.
result Implication of a Bernstein-type theorem for CMC surfaces in isotropic 3-space.
In this paper, we study factorable surfaces in a 3-dimensional isotropic space. We classify such surfaces with constant isotropic Gaussian (K) and mean curvature (H). We provide a non-existence result related with the surfaces satisfying H/K=const. Several examples are also illustrated.
In this paper, we study the rotational surfaces in the isotropic 3-space I^3. satisfying Weingarten conditions in terms of the relative curvature K (analogue of the Gaussian curvature) and the isotropic mean curvature H. In particular, we classify such surfaces of linear Weingarten type in I^3.
Study proves only origin-centered spheres solve certain curvature problems.
problem Proving uniqueness of solutions to curvature problems.
method Using the Heintze-Karcher inequality, the study proves the uniqueness of smooth, strictly convex solutions to a class of Minkowski type problems.
result Only origin-centered spheres solve isotropic and -Gaussian-Minkowski problems.
New blurring diffusion models bridge heat dissipation and denoising.
problem Developing a new generative modeling approach.
method Connecting blurring to Gaussian diffusion with non-isotropic noise.
result Proposed Blurring Diffusion Models offer the best of both Gaussian denoising and inverse heat dissipation.
Paper calculates KL divergence for isotropic Gaussian-Markov fields.
problem Measuring divergence between isotropic Gaussian-Markov fields.
method Derives closed-form KL divergence expressions.
result Develops new similarity measures in image processing.
The paper develops efficient algorithms for variational inference with mixtures of isotropic Gaussians.
problem Efficiently approximating multimodal Bayesian posteriors.
method Develops a variational framework and efficient algorithms for mixtures of isotropic Gaussians.
result The approach provides accurate approximations of multimodal Bayesian posteriors while being memory and computationally efficient.
The isotropic 3-space I^3 which is one of the Cayley--Klein spaces is obtained from the Euclidean space by substituting the usual Euclidean distance with the isotropic distance. In the present paper, we give several classifications on the surfaces in I^3 with the constant relative curvature (analogue of the Gaussian cu…
In this work, we are interested in the differential geometry of surfaces in simply isotropic and pseudo-isotropic spaces, which consists of the study of equipped with a degenerate metric such as . The investigation is…
We study invariant surfaces generated by one-parameter subgroups of simply and pseudo isotropic rigid motions. Basically, the simply and pseudo isotropic geometries are the study of a three-dimensional space equipped with a rank 2 metric of index zero and one, respectively. We show that the one-parameter subgroups of i…
Proposes Gaussian process priors on graph sets with geometric structure.
problem Defining Gaussian process priors on sets of graphs with geometric structure.
method Defines priors respecting graph geometric structure, analogous to Euclidean isotropic processes.
result Efficient computational technique for evaluating priors' kernels, making them usable in toolboxes.
Developed a new concept of isometric surfaces in isotropic space.
problem Missing notion of isometric deformations in isotropic space.
method Using Gauss' Theorema Egregium as a condition, developed a new concept.
result Natural analogues of Euclidean space isometries found in isotropic space.
We study learning problems in which the conditional distribution of the output given the input varies as a function of additional task variables. In varying-coefficient models with Gaussian process priors, a Gaussian process generates the functional relationship between the task variables and the parameters of this con…
A spectral mixture representation of isotropic kernels with application to random Fourier featurescs.LG
New spectral mixture representation for isotropic kernels simplifies random Fourier features.
problem Applying Random Fourier Features to complex kernels.
method Decompose isotropic kernels into scale mixtures of α-stable random vectors.
result Constructive spectral sampling formula for various kernels.
In this study, we deal with the local structure of curves and surfaces immersed in a pseudo-isotropic space I_{p}^{3} that is a particular Cayley-Klein space. We provide the formulas of curvature, torsion and Frenet trihedron in order for spacelike and timelike curves. The causal character of all admissible surfaces in…
In this paper, we study the problem of finding the affine factorable surfaces in a 3-dimensional isotropic space with prescribed Gaussian (K) and mean (H) curvature. Because the absolute figure two different types of these surfaces appear by permutation of coordinates. We firstly classify the affine factorable surfaces…
SGD learns neural networks with a complexity measure called leap.
problem Time complexity of SGD learning on neural networks.
method Introduced a complexity measure called leap, proved conjecture for Gaussian data, and showed saddle-to-saddle dynamics.
result Proved a conjecture about the time complexity of learning functions with low-dimensional support.
Study on Gaussian random fields' singularities on manifolds.
problem Understanding singularities of Gaussian random fields on manifolds.
method Computed expected values of singularities under various conditions.
result Explicit formulae for singularities under different constraints.
High-dimensional smoothing techniques struggle with robustness guarantees against various attacks.
problem Challenges in extending randomized smoothing to other attack models in high-dimensional space.
method Analysis of isotropic and generalized Gaussian smoothing distributions, proving bounds on certified robustness radii.
result Certifiable robustness radii decrease as $O(1/d^{ rac{1}{2} - rac{1}{p}})$ with dimension for .
A scalable factorized Gaussian process VAE for faster inference.
problem Inference bottlenecks in Gaussian process VAEs.
method Factorizes latent kernel across auxiliary features, leveraging independence.
result Significant speed-up in inference time (in theory and practice).
The paper proves uniqueness of solutions to curvature problems using various methods.
problem Proving uniqueness of solutions to anisotropic and isotropic curvature problems.
method Integral formulas by S. S. Chern and Simon's uniqueness result, along with new methods.
result The only smooth strictly convex solution to the isotropic curvature problem is an origin-centred sphere.
Gradient span algorithms show consistent progress in high dimensions.
problem Understanding consistent training progress in large machine learning models.
method Proving deterministic behavior of gradient span algorithms on Gaussian random functions.
result Gradient span algorithms have asymptotically deterministic behavior in high dimensions.
The paper examines Randers metrics with isotropic scalar curvature properties.
problem Characterizing Randers metrics with specific scalar curvature properties.
method Analyzes properties of Randers metrics with isotropic scalar curvature.
result Proves that Randers metrics with weakly isotropic scalar curvature have isotropic -curvature and are either Minkowskian or Riemannian.
Paper establishes a relation between Berwald scalar curvature and S-curvature.
problem Understanding the relationship between Finsler metrics' curvature properties.
method Proved conditions for isotropic Berwald scalar curvature and weakly isotropic S-curvature.
result Finsler metrics with isotropic Berwald scalar curvature have weakly isotropic S-curvature.
In this paper, we find a condition on -metrics under which the notions of isotropic S-curvature, weakly isotropic S-curvature and isotropic mean Berwald curvature are equivalent.
Study physical work done by isotropic vector forces along isotropic curves.
problem Investigate physical work done by isotropic vector forces.
method Analyze forces represented by isotropic vectors acting along isotropic curves on a manifold with specific metric structures.
result Calculate the work done by isotropic vector forces along isotropic curves.
Polynomial networks converge to Gaussian processes at a rate of O(n^(-1/2)).
problem Understanding the convergence rate of polynomial networks to Gaussian processes.
method Examined one-hidden-layer neural networks with random weights, focusing on polynomial activations and their convergence rate in the 2-Wasserstein metric.
result The rate of convergence for polynomial networks to Gaussian processes is $O(n^{- rac{1}{2}})$.
Study on InstaHide's security, linking to phase retrieval problem.
problem Security of InstaHide scheme for private dataset sharing.
method Design of a provable algorithm for private vector recovery.
result Private vectors can be recovered using synthetic vectors and public vectors.
Study isotropic Riemannian maps and helices along them.
problem Understanding Riemannian maps and their associated helices.
method Presented isotropic Riemannian maps and characterized helices along them.
result Characterization of helices along Riemannian maps.
Bayesian framework for sphere regression using Gaussian fields.
problem Nonparametric regression on the sphere with Gaussian priors.
method Isotropic Gaussian field priors, harmonic structure, exact posterior distributions, optimal spectral truncation, posterior contraction rates.
result Sharp posterior contraction rates for Gaussian priors with polynomially decaying angular power spectra.
Projection pursuit model improves Gaussian process regression for high-dimensional data.
problem Scalability issues with traditional Gaussian process models in high dimensions.
method Additive Gaussian process regression with dimension expansion and gradient descent.
result The proposed method approximates more complex functions and outperforms traditional models.
We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic planes. We specialize the plane axiom for Riemannian manifolds of indefinite met…
Stochastic gradient descent converges to universal limits in high dimensions.
problem Statistical tasks in high dimensions with specific data projections.
method Stochastic gradient descent applied to mixture distributions, proving universality of limits.
result The ODE limits are universal for mixtures of arbitrary product distributions.
While several papers have investigated computationally and statistically efficient methods for learning Gaussian mixtures, precise minimax bounds for their statistical performance as well as fundamental limits in high-dimensional settings are not well-understood. In this paper, we provide precise information theoretic …
Study isotropic curves on complex quadric with geometric relations.
problem Characterize isotropic curves on the complex quadric.
method Analyze geometric properties and relations of isotropic curves.
result Discovers relations between isotropic curves and surfaces in spaceforms.
In this paper, we construct a new class of Finsler manifolds called generalized isotropic Berwald manifolds which is an extension of the class of isotropic Berwald manifolds. We prove that every generalized isotropic Berwald manifold is a generalized Douglas-Weyl manifold. On a compact generalized isotropic Berwald man…
Constructs a moment map flow for isotropic maps on surfaces.
problem Understanding isotropic maps on surfaces and their properties.
method Develops a Kähler moment map geometry and a modified moment map flow.
result Polyhedral modified moment map flow induces a strong deformation retraction.
Novel Hilbert space Gaussian process improves sequential design accuracy and efficiency.
problem Efficiently implementing Gaussian process acquisition functions for expensive simulations.
method Proposed a truncated eigenbasis representation for closed-form evaluation of IMSE acquisition function.
result Significantly lower prediction error and reduced computation time compared to benchmarks.
Spinor representation in isotropic space via Laguerre geometry.
problem Representing conformal and constant mean curvature surfaces in isotropic space.
method Developing Laguerre geometry of isotropic space, defining spin transformations, and constructing Weierstrass and Kenmotsu representations.
result Explicit constructions of zero mean curvature and constant mean curvature surfaces.
The paper studies hanging chains and surfaces in degenerate geometries.
problem Investigating hanging chains and surfaces in simply isotropic plane and space.
method Characterizing catenaries and proving them as minimal surfaces in the simply isotropic space.
result The simply isotropic catenary is the generating curve of a minimal surface of revolution.
We create real-time geodesic rendering for non-isotropic geometries.
problem Challenging visualization of non-isotropic geometries.
method Novel methods for real-time native geodesic rendering.
result Methods can be applied to visualization, machine learning, and video games.
Paper shows isotropic - and -curvatures are equivalent in warped Finsler metrics.
problem Equivalence of non-Riemannian curvatures in warped product Finsler metrics.
method Study of warped product Finsler metrics.
result Isotropic - and -curvatures are equivalent for these metrics.
In this paper we will show that a Lagrangian, Lorentzian surface in a complex pseudo space form is pseudo-isotropic if and only if is minimal. Next we will obtain a complete classification of all Lagrangian, Lorentzian surfaces which are lightlike pseudo-isotropic but not pseudo-isot…
The paper studies Kropina metrics with a specific curvature property.
problem Characterizing Kropina metrics with isotropic scalar curvature.
method Tensor analysis to derive expressions and characterize metrics.
result Characterization of Kropina metrics with isotropic scalar curvature.
New approach uses isotropic geometry to solve Euclidean problems.
problem Solving systems of constraints in Euclidean geometry.
method Start with analogous problems in isotropic geometry to initialize optimization algorithms.
result Solutions in isotropic geometry provide insight and initialize Euclidean problem solutions.
Isotropic tori in revisitedmath.SG
We show that for , there are at least two exact isotropic -tori in which are not Hamiltonian isotopic in , even though they are smoothly isotopic as isotropic -tori. We apply this discovery to obtain more distinct non-exact isotropic tori in .
LeJEPA provides a scalable, theory-driven approach to self-supervised learning.
problem Lack of practical guidance and theory in JEPAs.
method Identified optimal Gaussian distribution and introduced SIGReg objective.
result LeJEPA achieves state-of-the-art performance with minimal hyperparameters and heuristics.