Study on nodal components of random band-limited functions on surfaces, finding a universal law.
problem Distribution of tangencies of nodal components to a vector field on surfaces.
method Analysis of random band-limited functions on smooth compact Riemannian surfaces with vector fields.
result The distribution of tangencies to a vector field on nodal components of random band-limited functions on surfaces follows a universal deterministic law.
Optimizes PDE-constrained LDDMM for efficient non-rigid registration.
problem Inexact Newton-Krylov optimization in PDE-constrained LDDMM leads to poor geodesic paths.
method Band-limited vector field parameterization to optimize computational complexity.
result Optimized method shows competitive performance with reduced memory load and computational time.
Band-limited training reduces resource usage without sacrificing accuracy.
problem Resource constraints in training Convolutional Neural Networks (CNNs).
method Artificially constraining the frequency spectra of convolutional filters during training.
result CNNs can leverage lower-frequency components effectively, reducing resource usage.
The paper improves confidence regions for band-limited functions using tighter norm bounds and majority voting.
problem Constructing reliable confidence regions for band-limited functions from noisy data.
method Improved norm bounds using Hoeffding's inequality and empirical Bernstein bound, majority voting to aggregate intervals.
result Confidence intervals retain their simultaneous coverage guarantee even when aggregated from random subsamples.
The paper creates nonparametric confidence bands for band-limited functions.
problem Estimating confidence bands for band-limited functions with finite samples and unknown noise.
method Uses Paley-Wiener reproducing kernel Hilbert spaces and gradient-perturbation methods.
result Non-asymptotic guarantees for confidence regions without assuming a parametric model.
We study the volume distribution of nodal domains of random band-limited functions on generic manifolds, and find that in the high energy limit a typical instance obeys a deterministic universal law, independent of the manifold. Some of the basic qualitative properties of this law, such as its support, monotonicity and…
We introduce a new GP kernel based on the sinc function for band-limited signals.
problem Designing covariance kernels for band-limited signals.
method Proposes a Gaussian process kernel with a power spectral density modeled by a rectangular function.
result The sinc kernel facilitates efficient signal processing applications like stereo modulation and band-pass filtering.
In this paper, a nonparametric maximum likelihood (ML) estimator for band-limited (BL) probability density functions (pdfs) is proposed. The BLML estimator is consistent and computationally efficient. To compute the BLML estimator, three approximate algorithms are presented: a binary quadratic programming (BQP) algorit…
Wiatowski and Bölcskei, 2015, proved that deformation stability and vertical translation invariance of deep convolutional neural network-based feature extractors are guaranteed by the network structure per se rather than the specific convolution kernels and non-linearities. While the translation invariance result appli…
This work uses sampling theory to analyze smoothness and error bounds of finite neural networks.
problem Analyzing the function space of finite neural networks and providing error bounds.
method Applying sampling theory to finite neural networks with non-expansive activation functions, considering both deterministic and random sampling.
result Novel error bounds for univariate neural networks under band-limited input assumption, highlighting the advantage of deterministic uniform sampling.
The paper improves nonparametric confidence bands for band-limited functions.
problem Constructing nonparametric simultaneous confidence bands with nonasymptotic and distribition-free guarantees.
method Based on Paley-Wiener reproducing kernel Hilbert spaces, the paper relaxes assumptions, improves noise estimation, and tightens constraints.
result Enhanced confidence bands with improved efficiency and tighter constraints.
Deep convolutional neural networks (CNNs) used in practice employ potentially hundreds of layers and 10,000s of nodes. Such network sizes entail significant computational complexity due to the large number of convolutions that need to be carried out; in addition, a large number of parameters needs to be learned and…
Band-limited SAC improves learning efficiency and stability in simulated environments.
problem Improving sample efficiency and stability in SAC algorithms.
method Artificially bandlimiting the target critic's spatial resolution using a convolutional filter.
result Bandlimited SAC outperforms classic twin-critic SAC in various Gym environments and is more stable.
Compression of Neural Networks (NN) has become a highly studied topic in recent years. The main reason for this is the demand for industrial scale usage of NNs such as deploying them on mobile devices, storing them efficiently, transmitting them via band-limited channels and most importantly doing inference at scale. I…
Paper transforms torse-forming vector fields into simpler forms.
problem Generalizing vector fields and their transformations.
method Present techniques to transform torse-forming vector fields into simpler cases.
result Concrete examples of transformations are provided.
The paper proves that certain modified conformal vector fields are trivial on compact and non-compact manifolds.
problem Proving triviality of modified conformal vector fields on Riemannian manifolds.
method Analyzing properties of homothetic, conformal, and gradient vector fields on compact and non-compact manifolds.
result Established conditions under which m-modified conformal vector fields are trivial. This paper presents a bias-variance tradeoff of graph Laplacian regularizer, which is widely used in graph signal processing and semi-supervised learning tasks. The scaling law of the optimal regularization parameter is specified in terms of the spectral graph properties and a novel signal-to-noise ratio parameter, whi…
Characterizes winding of braided vector fields in tubular domains.
problem Understanding the topology of braided vector fields in complex domains.
method Defines field line winding as a measure of entanglement, proving its uniqueness in classifying vector field topology.
result Field line winding uniquely classifies the topology of braided vector fields.
Every smooth vector field is a combination of gradient fields.
problem Expressing arbitrary smooth vector fields as combinations of gradient fields.
method Proving every smooth vector field can be written as a finite linear combination of iterated Lie brackets of gradient vector fields.
result Every smooth vector field is a combination of gradient fields.
The position vector field x is the most elementary and natural geometric object on a Euclidean submanifold M. The position vector field plays very important roles in mathematics as well as in physics. Similarly, the tangential component x^T of the position vector field is the most natural vector field tangent to the …
Conformal vector fields on LCP manifolds are orthogonal and Killing.
problem Understanding conformal vector fields on specific geometric manifolds.
method Analyzing properties of conformal vector fields on compact locally conformally product manifolds.
result Conformal vector fields are orthogonal to the flat distribution and Killing.
For a submanifold M in a Euclidean space, the tangential component x^T of the position vector field x of M is the most natural vector field tangent to the Euclidean submanifold, called the canonical vector field of M. In this article, first we prove that the canonical vector field of every Euclidean submanifold is alwa…
Study on vector fields on Lie groups reveals surprising algebraic coincidences.
problem Characterizing vector fields on Lie groups with Riemannian metrics.
method Algebraic and geometric analysis of left-invariant vector fields on nilpotent Lie groups.
result Spaces of Killing, one-harmonic, and conformal vector fields coincide with the center of the Lie algebra on nilpotent Lie groups.
Study biharmonic vector fields and unit vector fields on Riemannian manifolds.
problem Determine the equivalence of biharmonicity and harmonicity for vector fields and unit vector fields on Riemannian manifolds.
method Analyze biharmonic vector fields and unit vector fields on (M,g) with pseudo-Riemannian g-natural metrics on TM and T1M. result Contrary to Sasaki metric, biharmonicity and harmonicity are not equivalent for large classes of g-natural metrics on TM. This short report establishes some basic properties of smooth vector fields on product manifolds. The main results are: (i) On a product manifold there always exists a direct sum decomposition into horizontal and vertical vector fields. (ii) Horizontal and vertical vector fields are naturally isomorphic to smooth famil…
Defines quaternionic k-vector fields on quaternionic Kähler manifolds.
problem No specific problem stated; focuses on definition and properties.
method Introduced a modified Dirac operator to define quaternionic k-vector fields.
result Calculated the dimension of quaternionic k-vector fields on HPn. Study on generalized derivations in polynomial vector fields Lie algebras.
problem Understanding generalized derivations in specific Lie sub-algebras of polynomial vector fields.
method Analysis of Lie sub-algebras containing constant and Euler vector fields, under specified conditions.
result Characterization of generalized derivations in the studied Lie sub-algebras.
Characterizes affine vector fields on Finsler manifolds with rigidity results.
problem Understanding affine vector fields on Finsler manifolds.
method Utilizing the Jacobi type equation and spray characterization, proving rigidity theorems.
result Rigidity theorems for affine vector fields on Finsler manifolds with non-positive total Ricci curvature.
Study on Einstein solitons with specific vector fields and their properties.
problem Characterizing Einstein solitons with gradient, solenoidal, or concircular vector fields.
method Explicitly express the function λ by gradient vector field V and deduce geometric properties under certain curvature conditions.
result Explicit expressions for λ and geometric properties of Einstein solitons.
Investigates point spectra of vector fields and their properties.
problem Understanding the point spectra of vector fields.
method Define and study point spectra, prove properties under isometries, and analyze compactly supported fields.
result Point spectra are well-behaved under isometries and trivial for compactly supported fields.
Study describes lcK structures on Vaisman-type manifolds with holomorphic Lee vector field.
problem Characterizing lcK structures with holomorphic Lee vector field on Vaisman-type manifolds.
method Complete description through potential analysis and vector field properties.
result Examples of lcK structures with non-homothetic Lee vector field.
We consider four dimensional lie groups equipped with left invariant Lorentzian Einstein metrics, and determine the harmonicity properties of vector fields on these spaces. In some cases, all these vector fields are critical points for the energy functional restricted to vector fields. We also classify vector fields de…
Vector fields invariant under Lie group action are finitely generated by polynomial fields.
problem Understanding invariant vector fields under Lie group actions.
method Analyzing the module of smooth vector fields invariant under a linear action of a compact Lie group.
result The module of invariant vector fields is finitely generated by polynomial fields.
Examining singularities of commuting vector fields on submanifolds.
problem Understanding singularities of commuting vector fields on submanifolds.
method Analyzing real submanifolds within Kähler manifolds.
result Characterized singularities of commuting vector fields.
The paper proves non-existence of torqued and anti-torqued vector fields on hyperbolic spaces.
problem Existence of torqued and anti-torqued vector fields on hyperbolic spaces.
method Analyzing the properties of conformal scalar functions and their impact on the existence of vector fields.
result Non-existence of proper torqued and anti-torqued vector fields on hyperbolic spaces.
We introduce G_2-vector fields, Rochesterian 1-forms and Rochesterian vector fields on manifolds with a closed G_2-structure as analogues of symplectic vector fields, Hamiltonian functions and Hamiltonian vector fields respectively, and we show that the spaces of G_2-vector fields and of Rochesterian vector fields are …
Enhances Hamiltonian systems stability through generalized double bracket vector fields.
problem Stabilizing already stable points in Hamiltonian systems.
method Generalized double bracket vector fields on Poisson manifolds with pseudo-Riemannian metrics.
result Enhanced equilibria stability through dissipation terms.
Study classifies harmonic vector fields on 3-manifolds.
problem Classifying harmonic unit vector fields on 3-manifolds.
method Investigates under mild curvature assumptions, classifying vector fields and manifolds.
result Classifies both vector fields and manifolds supporting them.
Study proves conformal vector fields on certain Finsler manifolds are Killing fields.
problem Characterizing conformal vector fields on compact homogeneous Finsler manifolds.
method Analyzes properties of conformal vector fields and homogeneous Finsler metrics.
result Conformal vector fields on compact homogeneous Finsler manifolds are Killing fields.
Indices of vector fields and 1-forms studied for singular varieties and actions.
problem Understanding indices of vector fields and 1-forms in various contexts.
method Generalization to singular varieties and actions of finite groups.
result New insights into indices of vector fields and 1-forms.
Harmonic basis vector fields on surfaces
problem Parameterizing surfaces with harmonic vector fields
method Introducing harmonic basis vector fields and deriving conditions for their existence
result Classifying parameterizations of surfaces with harmonic basis vector fields
The paper explores how vector fields relate to volume in geometric contexts.
problem Existence of nondiffeomorphic contact forms with identical Reeb vector fields.
method Analyzes geodesible vector fields and their associated Euler classes, applying topological and geometric theorems.
result Proves the Gauss-Bonnet and Poincaré-Hopf theorems for 2D orbifolds using geodesible vector fields.
A Ricci soliton (M,g,v,λ) on a Riemannian manifold (M,g) is said to have concurrent potential field if its potential field v is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were studied recently in \cite{CD2}. The most important concurrent vector field is …
Study shows limitations of Lie bracket commutation for nonsmooth vector fields.
problem Limitations of Lie bracket commutation for nonsmooth vector fields.
method Analysis of nonsmooth vector fields, focusing on commutation of flows and Lie bracket conditions.
result Lie bracket commutation cannot be extended to general a.e. differentiable vector fields, but holds for certain Sobolev regular fields.
Study on almost Riemann solitons with gradient or torse-forming vector fields.
problem Characterizing almost Riemann solitons with specific vector fields.
method Using Bochner formula and properties of gradient and torse-forming vector fields.
result Explicit expressions for the soliton function λ under gradient and torse-forming conditions. Discuss various definitions of vector fields on manifolds leading to Lie algebras.
problem Defining vector fields on convenient manifolds.
method Various definitions of vector fields and their equivalence.
result Equivalent definitions of vector fields on manifolds.
In this paper, we define a certain "proportional volume property" for an unit vector field on a spherical domain in S3. We prove that the volume of these vector fields has an absolute minimum and this value is equal to the volume of the Hopf vector field. Some examples of such vector fields are given. We also study the…
The study classifies spaces with specific conformal vector fields.
problem Characterizing closed vacuum static spaces with non-Killing conformal vector fields.
method Provided characterizations and established an identity involving the characteristic function.
result Derived a rigidity theorem and classified spaces with the vector field.