In this article we address a number of features of the moduli space of spherical metrics on connected, compact, orientable surfaces with conical singularities of assigned angles, such as its non-emptiness and connectedness. We also consider some features of the forgetful map from the above moduli space of spherical sur…
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48 results for “spherical features”
Revisits Gaussian process model with spherical harmonics for scalable deep learning.
problem Scaling Gaussian process models to large input dimensions with high frequency learning.
method Introduces new kernels related to deep models, variational learning of spherical harmonic phases, and sparseness in eigenbasis.
result Enables scaling to larger input dimensions and learning of high frequency variations.
Enhances Gaussian processes with spherical features for better scalability and flexibility.
problem Lack of representation learning in Gaussian processes compared to deep neural networks.
method Introduces spherical inter-domain features to improve GP approximation and scalability.
result The method alleviates limitations and improves scalability compared to alternative strategies.
New Fourier features improve high-precision approximation in large-scale problems.
problem Designing scalable, high-precision Fourier features for large-scale kernel methods.
method Introducing a new family of quadrature rules that accurately approximate the Gaussian measure in higher dimensions.
result Improved approximation bounds with new Fourier features.
Study local features of decorated representation spaces for spherical surfaces.
problem Local structure of moduli space of spherical surfaces with conical points.
method Analysis of decorated representation spaces of fundamental groups in SU(2).
result Smooth locus of decorated representation spaces is dense and connected.
A new GP model uses spherical harmonics for faster inference.
problem Efficiently fitting large datasets with Gaussian processes.
method Sparse Gaussian processes with spherical harmonic features.
result Significant speed-up in inference for large datasets.
Recovering the number of clusters in data sets with noise features using feature rescaling factorsstat.ML
In this paper we introduce three methods for re-scaling data sets aiming at improving the likelihood of clustering validity indexes to return the true number of spherical Gaussian clusters with additional noise features. Our method obtains feature re-scaling factors taking into account the structure of a given data set…
Paper proposes SMFN for high-res spherical video super-resolution.
problem Super-resolution of 360-degree panoramic videos is expensive and challenging.
method Deformable convolutions, mixed attention mechanism, dual learning strategy, weighted mean square error loss function.
result The proposed SMFN method improves super-resolution of equatorial regions in 360-degree videos.
A new geometric model for V1 hypercolumns combines symplectic and spherical models.
problem Understanding the structure of V1 hypercolumns in the visual cortex.
method A differential geometric model based on conformal geometry.
result Combines features of symplectic and spherical models of hypercolumns.
New analysis tightens memory capacity of Hopfield models using spherical codes.
problem Optimizing memory capacity in modern Hopfield models and Kernelized Hopfield Models.
method Connecting Hopfield models to spherical codes in information theory, establishing an optimal capacity bound and a sub-linear algorithm.
result First tight and optimal asymptotic memory capacity for modern Hopfield models, matching known lower bounds.
We consider the problem of clustering data points in high dimensions, i.e. when the number of data points may be much smaller than the number of dimensions. Specifically, we consider a Gaussian mixture model (GMM) with non-spherical Gaussian components, where the clusters are distinguished by only a few relevant dimens…
Derives TAP approximation for Bayesian linear regression.
problem Log-normalizing constant of posterior distribution in high-dimensional linear regression.
method Variational representation and Thouless-Anderson-Palmer approximation.
result Proves TAP approximation for spherical prior in proportional asymptotic regime.
We show how to compute the circular area invariant of planar curves, and the spherical volume invariant of surfaces, in terms of line and surface integrals, respectively. We use the Divergence Theorem to express the area and volume integrals as line and surface integrals, respectively, against particular kernels; our r…
Paper relaxes symmetry conditions for universal feature selection in noisy data.
problem Feature selection in noisy data with weak symmetry.
method Developed a universal feature selection framework using singular value decomposition of canonical dependence matrix.
result Selected features achieve asymptotically optimal error exponents up to a residual term.
Two-layer NN with channel attention learns low-degree spherical polynomials efficiently.
problem Learning low-degree spherical polynomials with over-parameterized neural networks.
method Two-layer neural network with channel attention, vanilla gradient descent, learnable channel selection.
result Minimally improved sample complexity of $n \asymp Θ(d^{\ell_0}/\eps)$ for learning low-degree polynomials.
KATA improves associative recall by optimizing feature maps derived from nonnegative attention weights.
problem Linear attention's poor performance on associative recall tasks.
method Formulates attention recall as a spherical-packing problem and introduces Kernelized Linear Attention Activations (KATA).
result KATA features offer a favorable capacity-interference tradeoff, enabling efficient associative recall.
This paper deals with the notion of quadratic differential in spherical CR geometry (or more generally on strictly pseudoconvex CR manifolds). We get to this notion by studying a splitting of Rumin complex and discuss its first features such as trajectories and length. We also define several differential operators on q…
The paper verifies deep neural networks' ability to approximate functions on spheres.
problem Theoretical verification of deep neural networks' performance on spherical functions.
method Spherical analysis using reproducing kernels and convolutional factorizations.
result Rates of uniform approximation for functions in Sobolev spaces and additive ridge forms.
FHDMs achieve optimal convergence in spherically supported data.
problem Statistical convergence properties of FHDMs for spherical data.
method FHDMs leverage random generation time and Doob's h-transform to optimize convergence rate.
result Achieve minimax optimal convergence rate in total variation for spherically supported Sobolev smooth data.
Recent work by Cohen \emph{et al.} has achieved state-of-the-art results for learning spherical images in a rotation invariant way by using ideas from group representation theory and noncommutative harmonic analysis. In this paper we propose a generalization of this work that generally exhibits improved performace, but…
Spherical Knot Mosaicsmath.GT
Paper introduces spherical knot mosaics for knot and link invariants.
problem Representing knots on a sphere with tiles.
method Tiling a 2-sphere with 11 knot mosaic tiles to define new invariants.
result New knot invariants derived from spherical mosaic tiling.
Support Vector Data Description (SVDD) is a machine-learning technique used for single class classification and outlier detection. SVDD formulation with kernel function provides a flexible boundary around data. The value of kernel function parameters affects the nature of the data boundary. For example, it is observed …
Study geodesics on spherical polyhedra, estimating their number.
problem Counting simple closed geodesics on spherical polyhedra.
method Examined regular spherical octahedra, cubes, and tetrahedra.
result Estimated the number of simple closed geodesics on spherical polyhedra.
The abstract proves spherical surface decompositions with conical singularities.
problem Decomposing surfaces with spherical metrics and conical singularities.
method Geometric triangulations and irreducible components of standard shapes.
result Spherical polygons, including half-spherical concave polygons, can be arbitrarily complicated.
For the n-dimensional spherical pedal curve with respect to an n-dimensional spherical unit speed curve and a given point , we define the spherical orthotomic curve of relative to the point , and classify singularities of spherical orthotomic curves.
Study spherical curves with curvature dependent on distance to a great circle.
problem Understanding spherical curves with curvature dependent on distance to a great circle.
method Introducing spherical angular momentum, characterizing known curves, finding new families, and obtaining arc length parametrizations.
result New families of spherical curves with intrinsic equations in elementary or Jacobi elliptic functions.
Study calculates eigenvalues and eigenfunctions for spherical triangles and finds fundamental gap behavior.
problem Understanding eigenvalues and gaps in spherical triangles.
method Explicit computation of Dirichlet eigenvalues and eigenfunctions for spherical lunes and triangles.
result Fundamental gap of spherical triangles increases as the angle of the lune decreases.
New findings show fundamental group is not audible in spherical space forms.
problem Isospectral spherical space forms with non-cyclic fundamental groups.
method Revisited and found new examples of spherical space forms.
result Fundamental group is not audible among spherical space forms.
Existence and uniqueness of spherical helicoidal surfaces in 3-sphere via spherical curves.
problem Existence and uniqueness of spherical helicoidal surfaces in 3-sphere.
method Continuous function of distance to axis, spherical angular momentum of spherical curves.
result Existence and uniqueness theorem for spherical helicoidal surfaces in 3-sphere.
The reductivity of a spherical curve represents how reduced the spherical curve is. It is unknown if there exists a spherical curve whose reductivity is four. In this paper we give an unavoidable set for spherical curves with reductivity four by considering 4-gons.
The paper improves inequalities for nearly spherical sets using quermassintegrals.
problem Improving inequalities for nearly spherical sets.
method Establishing quantitative Alexandrov-Fenchel inequalities for quermassintegrals.
result Lower bounds on the -isoperimetric deficit found using spherical deviation and asymmetry.
The reductivity of a spherical curve is the minimal number of a local transformation called an inverse-half-twisted splice required to obtain a reducible spherical curve from the spherical curve. It is unknown if there exists a spherical curve whose reductivity is four. In this paper, an unavoidable set of configuratio…
We show that we can obtain a reducible spherical curve from any non-trivial spherical curve by four or less inverse-half-twisted splices, i.e., the reductivity, which represents how reduced a spherical curve is, is four or less. We also discuss unavoidable sets of tangles for spherical curves.
The paper classifies Landsberg spherically symmetric Finsler metrics in various dimensions.
problem Investigating compatibility conditions on spherically symmetric Finsler metrics.
method Using the inverse problem of calculus of variations, the paper focuses on Landsberg and Berwald types.
result All Landsberg spherically symmetric manifolds in higher dimensions are either Riemannian or have specific geodesic spray formulas.
For a convex body on the Euclidean unit sphere the spherical convex floating body is introduced. The asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated. This gives rise to a new spherical area measure, the floating area. Remarkably, this floating area…
Study on spherical CR manifolds with non-trivial Chern classes.
problem Understanding Chern classes in spherical CR manifolds.
method Construction and proof of constraints on Chern classes.
result Topological obstruction to spherical CR structures on contact manifolds.
PGF kernels analyze spherical data using generalized RBF kernels.
problem Analysis of spherical data.
method Introduced PGF kernels and a semi-parametric learning algorithm.
result PGF kernels generalize RBF kernels for spherical data.
The paper examines properties of spherical Finsler metrics, proving their semi-C-reducibility and conditions for vanishing mean stretch curvature.
problem Analyzing curvature properties of spherical Finsler metrics.
method Proving semi-C-reducibility and finding conditions for vanishing mean stretch curvature.
result Conditions for a general spherically symmetric Finsler metric to have vanishing mean stretch curvature.
Classifies involutions on spherical 3-manifolds.
problem Classifying involutions on spherical 3-manifolds.
method Geometric approach to conjugacy classification.
result Insights into topological properties of involutions.
The paper classifies spherically symmetric Finsler metrics as Berwaldian or Riemannian.
problem Classifying spherically symmetric Finsler metrics.
method Proving all spherically symmetric Landsberg surfaces are Berwaldian and modifying the classification of spherically symmetric Finsler metrics.
result All Berwald spherically symmetric metrics of dimension are Riemannian or given by a specific formula.
Paper develops invariants for spherical curves using chord diagrams.
problem Developing invariants for spherical curves under local moves.
method Using based chord diagrams and local moves from Reidemeister moves.
result Invariants include both classical and new spherical curve invariants.
In this paper we consider the spherical slant helices in . More- over, we show how could be obtained to a spherical slant helix and we give some spherical slant helix examples in Euclidean 3-space.
The paper characterizes spherically symmetric metrics with scalar curvature.
problem Characterizing spherically symmetric metrics with scalar curvature.
method Established a curvature compatibility condition on spherically symmetric Finsler metrics and constructed a Berwald frame.
result Characterized spherically symmetric metrics with scalar curvature.
The study proves spherical polygon analogs of curve theorems, finding bounds on intersections and inflections.
problem Finding bounds on intersections and inflections for spherical polygons.
method Adapting smooth curve theorems to spherical polygons using discrete tools.
result Proves discrete analogs of four-vertex theorems for spherical polygons.
New formula for spherical polygon area via prequantization.
problem Traditional area formula for spherical polygons requires measuring angles.
method Uses prequantization to create a new formula that doesn't require angle measurement.
result New formula applicable to a wider range of degenerate curves and polygons.
Euler explored spherical geometry using trigonometric formulae and solid geometry methods.
problem Developing trigonometric formulae for spherical geometry.
method Used calculus of variations and classical solid geometry methods.
result Established trigonometric formulae and computed spherical triangle areas.
A spherical polyhedron surface is a triangulated surface obtained by isometric gluing of spherical triangles. For instance, the boundary of a generic convex polytope in the 3-sphere is a spherical polyhedron surface. This paper investigates these surfaces from the point of view of inner angles. A rigidity result is obt…
We propose a novel model for generating graphs similar to a given example graph. Unlike standard approaches that compute features of graphs in Euclidean space, our approach obtains features on a surface of a hypersphere. We then utilize a von Mises-Fisher distribution, an exponential family distribution on the surface …