Study examines structures on special hyperspheres in a 6-sphere.
problem Characterizing structures on hyperspheres in nearly Kaehler 6-sphere.
method Analyzes two types of almost contact metric structures.
result Discovers a one-parameter family of totally umbilical hyperspheres.
We consider non-degenerate centro-affine hypersurface immersions in R^n whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a bijective correspondence between homothetic families of proper affine hyperspheres with center in the origin and with parallel cubic form, …
Paper presents exact heat kernel on hypersphere for SVM improvements.
problem Improving SVM performance with non-Euclidean feature spaces.
method Higher order parametrix expansion of hyperspherical heat kernel.
result Exact kernel often shows superior performance in SVM applications.
Paper optimizes hyperspherical prototypes for better class separation.
problem Previous HPL approaches either lack principled optimisation or are limited to one latent dimension.
method Develops a principled optimisation procedure and uses linear block codes to create well-separated prototypes in various dimensions.
result Optimal prototype placement is characterized with achievable and converse bounds, showing near-optimality.
The paper reviews statistical models for high-dimensional normalized vectors.
problem Handling directional data in machine learning.
method Review of mathematical models for normalized vectors on hypersphere and real projective plane.
result Common models and technical aspects are discussed.
A new reservoir computing approach on the hypersphere surpasses memory limits.
problem Sequence learning and time series prediction problems.
method Random fixed-weight RNN on the unit hypersphere, removing non-linear activation.
result Memory capacity exceeds reservoir dimensionality, surpassing typical ESN limits.
New minimal hypersphere found in 4-sphere solving Bernstein problem.
problem Spherical Bernstein problem in S 4 \mathbb{S}^4 S 4 method Equivariant min-max theory for G G G -invariant minimal hypersurfaces result Construction of embedded non-equatorial minimal hypersphere
A new loss function HUG decouples and generalizes neural collapse.
problem Neural collapse limits in deep learning models.
method Hyperspherical uniformity gap (HUG) as a unified framework.
result HUG decouples and generalizes neural collapse, improving model flexibility and robustness.
In this note we prove that a constant mean curvature surface is proper-biharmonic in the unit Euclidean sphere S 4 \mathbb{S}^4 S 4 if and only if it is minimal in a hypersphere S 3 ( 1 2 ) \mathbb{S}^3(\frac{1}{\sqrt{2}}) S 3 ( 2 1 ) .
T-PSDA improves speaker recognition accuracy on toroidal submanifolds.
problem Improving speaker recognition accuracy on hypersphere embeddings.
method Extends PSDA to model within and between-speaker variabilities in toroidal submanifolds of the hypersphere.
result T-PSDA achieves accuracy on par with cosine scoring on VoxCeleb and large accuracy gains on NIST SRE'21.
A new method for speaker recognition on hyperspheres improves on PLDA's limitations.
problem Improving speaker recognition on hyperspheres with PLDA's limitations.
method Probabilistic Spherical Discriminant Analysis (PSDA) using Von Mises-Fisher distributions.
result PSDA scores are closed-form and can handle various trials, improving over PLDA.
The paper studies canal hypersurfaces in Lorentz-Minkowski 4-space.
problem Characterizing canal hypersurfaces formed by pseudo null, partially null, and null curves.
method Obtained parametric expressions and geometric invariants of canal hypersurfaces.
result Characterizations of tubular hypersurfaces in E 1 4 E^4_1 E 1 4 . A new geometric perceptron model improves 3D shape classification.
problem Challenges in geometric tasks involving point clouds using machine learning.
method Introduces multilayer geometric perceptron (MLGP) with geometric neurons.
result MLGP outperforms vanilla MLP in 3D shape classification and noise resistance.
This paper uses a geometric approach to understand how normalization layers affect neural network optimization.
problem Understanding the effect of normalization layers on optimization in neural networks.
method Introduces a spherical framework to study optimization dynamics of neural networks with normalization layers from a geometric perspective.
result Derives the first effective learning rate expression of Adam and shows that SGD with NLs is equivalent to a constrained variant of Adam.
The paper studies canal hypersurfaces in Lorentz-Minkowski 4-space.
problem Characterizing canal hypersurfaces in Lorentz-Minkowski 4-space.
method Analyzing geometric invariants and conditions for canal hypersurfaces.
result Characterizations of canal hypersurfaces, including flatness and minimality conditions.
The paper classifies J ~ \widetilde{J} J -tangent affine hyperspheres in arbitrary dimensions.
problem Classifying J ~ \widetilde{J} J -tangent affine hyperspheres. method Analyzing the canonical para-complex structure and involutive distribution.
result Classification of J ~ \widetilde{J} J -tangent affine hyperspheres in arbitrary dimensions. The study classifies product affine hyperspheres in R^n+1.
problem Locally strongly convex affine hyperspheres in R^(n+1).
method Classification based on product structure and constant sectional curvatures.
result Complete classification of product affine hyperspheres established.
We describe extrinsic hyperspheres and totally geodesic hypersurfaces in manifolds with special holonomy. In particular we prove the nonexistence of extrinsic hyperspheres in quaternion-Kaehler manifolds. We develop a new approach to extrinsic hyperspheres based on the classification of special Killing forms.
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.
We show that if a compact hypersurface M ⊂ R n + 1 M \subset \mathbb{R}^{n+1} M ⊂ R n + 1 , n ≥ 3 n \geq3 n ≥ 3 , admits a non zero Killing vector field X X X of constant length then n n n is even and M M M is diffeomorphic to the unit hypersphere of R n + 1 \mathbb{R}^{n+1} R n + 1 . Actually, we show that M M M is a complex ellipsoid in C N = R n + 1 \mathbb{C}^{N} = \mathbb{R}^{n+1} C N = R n + 1 .…
Prototype networks on hyperspheres improve classification and regression.
problem Improving classification and regression performance.
method Using hyperspherical prototypes for classification and regression, optimizing prototypes through data-independent margin separation.
result Hyperspherical prototype networks outperform other methods in classification, regression, and their combination.
Constructs hyperspheres with prescribed mean curvature in Euclidean space.
problem Creating hyperspheres with a specific curvature in Euclidean space.
method Constructs families of smooth functions to fill Euclidean space with hyperspheres of prescribed mean curvature.
result Euclidean space can be filled with hyperspheres of prescribed mean curvature.
In this paper, we explicitly construct the Calabi composition of multiple affine hyperspheres possibly including some points viewing as 0-dimensional hypersheres. Then we compute all the basic affine invariants of the composed affine hyperspheres, proving that the composed affine hypersphere is symmetric one if and onl…
A new VAE model captures hyperspherical data better than the standard Gaussian model.
problem Standard VAEs fail to model data with a latent hyperspherical structure.
method Proposes using a von Mises-Fisher (vMF) distribution for the latent space, leading to a hyperspherical latent space.
result Hyperspherical VAEs outperform standard VAEs in capturing data with a hyperspherical latent structure.
The study characterizes kernel spaces on hyperspheres, impacting cubature algorithms.
problem Characterizing kernel spaces on hyperspheres for cubature algorithms.
method Characterization of Sobolev spaces and reproducing kernel Hilbert spaces over hyperspheres.
result Direct consequences for kernel cubature and worst-case error rates.
Study on hyperspheres in 4-spaces as special Riemannian manifolds.
problem Characterizing hyperspheres in Euclidean and Minkowski 4-spaces as specific Riemannian manifolds.
method Constructing and studying hyperspheres in 4-dimensional spaces (Euclidean and pseudo-Euclidean) as almost paracontact almost paracomplex Riemannian manifolds.
result Characterization and geometric properties of these manifolds.
SphereConv improves deep learning by learning angular representations on hyperspheres.
problem Challenges in training deep CNNs due to increased depth and larger parameter space.
method Introduces hyperspherical convolution (SphereConv) and deep hyperspherical convolution networks (SphereNet) to learn angular representations on hyperspheres.
result SphereNet effectively encodes discriminative representation and alleviates training difficulty.
Small hypersphere is unstable in both 4-harmonic and ES-4-harmonic settings.
problem Stability of small hypersphere in 4-harmonic and ES-4-harmonic settings.
method Analysis of normal variations of small hypersphere.
result Normal index of small hypersphere is one in both 4-harmonic and ES-4-harmonic settings.
A holomorphic representation formula for special parabolic hyperspheres is given.
Upper estimate of Dirac eigenvalue linked to hyperspherical radius.
problem Estimating the smallest eigenvalue of the Dirac operator.
method Proved an upper estimate of the smallest eigenvalue in terms of hyperspherical radius.
result Combining with known lower estimates, geometric consequences are derived.
This thesis models and approximates pose distributions in robot perception using quaternion and Gaussian methods.
problem Modeling and approximating probability distributions of poses in robot perception.
method Uses dual quaternions, unit quaternions, and Gaussian distributions to represent and approximate pose distributions.
result A framework for probabilistic modeling of poses in S 3 i m e s R 3 S_3 imes \mathbb{R}^3 S 3 im es R 3 that approximates various probability distributions. Horospheres, hyperspheres, and hyperplanes in hyperbolic spaces are rigid in terms of mean curvature.
problem Proving rigidity of geometric shapes in hyperbolic spaces.
method Analyzing perturbations of horospheres, hyperspheres, and hyperplanes to show they cannot increase their mean curvature.
result Horospheres, hyperspheres, and hyperplanes in hyperbolic spaces H n are rigid in terms of mean curvature.
The paper finds conditions for biharmonic orbits in symmetric spaces.
problem Conditions for biharmonic orbits in symmetric spaces.
method Analyzes isotropy representations and biharmonic submanifolds in hyperspheres.
result Necessary and sufficient conditions for biharmonic orbits in symmetric spaces.
Researchers enhance hyperspherical latent representations for higher-dimensional data.
problem Limited expressivity of hyperspherical vMF distribution in high dimensions.
method Use a product-space to extend hyperspherical parameterizations to higher dimensions.
result Improved results on image datasets compared to traditional methods.
Formulates integrals for hypersphere arrangements using cohomology and Cayley-Menger determinants.
problem Formulating integrals for hypersphere arrangements.
method Using cohomology and Cayley-Menger determinants.
result Explicit representation and variational formula of integrals.
This work optimizes alignment and uniformity of features on a hypersphere for better downstream performance.
problem Improving the performance of contrastive representation learning.
method Identifying and optimizing alignment and uniformity of features on a hypersphere.
result Directly optimizing alignment and uniformity leads to comparable or better performance than contrastive learning.
The paper proves volume bounds for specific hyperspheres in Riemannian manifolds.
problem Bounding the volume of hyperspheres in Riemannian manifolds.
method Analyzing mean curvature and curvature properties of hyperspheres in Riemannian manifolds.
result The volume of certain hyperspheres is bounded by 8 π 2 / 3 8\pi^2/3 8 π 2 /3 . We show how in many cases the algebraic number of immersed hyperspheres of constant (and prescribed) curvature may be related to the Euler Characteristic of the ambient space.
Researchers prove constant mean curvature graphs in hyperbolic 3-space for specific domains.
problem Existence of hyperbolic Killing graphs with constant mean curvature in exterior domains.
method Existence proof using CMC graphs and Killing vector fields.
result Existence of hyperbolic Killing graphs of constant mean curvature H in exterior domains.
We show that the natural S^1-bundle over a projective special Kaehler manifold carries the geometry of a proper affine hypersphere endowed with a Sasakian structure. The construction generalizes the geometry of the Hopf-fibration $\Sr^{2n+1} \longrightarrow \CP^n$ in the context of projective special Kaehler manifolds.…
Proposes MHE to reduce neural network redundancy, improving performance.
problem Redundancy in neural networks hinders generalization and computation.
method Inspired by Thomson's problem, MHE minimizes energy to regularize neural networks.
result MHE improves performance on various challenging tasks.
An affine hypersurface M is said to admit a pointwise symmetry, if there exists a subgroup G of Aut(T_p M) for all p in M, which preserves (pointwise) the affine metric h, the difference tensor K and the affine shape operator S. Here, we consider 3-dimensional indefinite affine hyperspheres, i.e. S= H Id (and thus S is…
Paper proposes efficient method to calculate Fisher-Bingham distribution normalizing constant.
problem Efficiently calculating the normalizing constant of Fisher-Bingham distributions.
method Numerical integration with continuous Euler transform to Fourier-type integral representation.
result The method is fast and accurate, applicable to high-dimensional distributions.
The study characterizes rectifying curves in n-dimensional space.
problem Understanding rectifying curves in arbitrary dimensions.
method Characterization through various conditions and constructions.
result Different ways to characterize rectifying curves in n-dimensional Euclidean space.
The study characterizes quadrics among affine hyperspheres based on centroid collinearity of sections.
problem Characterizing quadrics among affine hyperspheres based on section centroid collinearity.
method Extending Meyer and Reisner's theorem to unbounded convex sets and identifying additional assumptions.
result Ellipsoids, paraboloids, and one sheet of a two-sheeted hyperboloid are the only quadrics satisfying the centroid collinearity condition.
Classifies differential operators on spheres and hyperspheres.
problem Classifying conformally covariant differential operators between differential forms.
method Analyzing restriction of principal series representations of Lie group O(n+1,1).
result Explicit formulæ and factorization identities for matrix-valued operators.
We introduce and study the equiaffine symmetric {\bf hyperspheres}. For the first step we consider the locally strongly convex ones. In fact, by the idea used by Naitoh, we provide in this paper a direct proof of the complete classification for those affine symmetric hyperspheres. Then, via an earlier result of the fir…
This work studies the chord length distribution, in the case where both ends lie on a N N N -dimensional hypersphere ( N ≥ 2 N \geq 2 N ≥ 2 ). Actually, after connecting this distribution to the recently estimated surface of a hyperspherical cap \cite{SLi11}, closed-form expressions of both the probability density function and the cu…