Study biharmonic functions on vector bundles with spherical symmetry.
problem Investigate biharmonic functions on vector bundles with spherically symmetric metrics.
method Analyze vertical lifts and radial functions of functions on vector bundle manifolds.
result Construct an infinite two-parameter family of proper biharmonic functions.
We approximate derivatives of functions on manifolds by embedding them and applying vector-valued operators.
problem Derivatives of manifold-valued functions are harder to approximate than vector-valued functions.
method Embed the manifold into a higher space, approximate the derivative of the vector-valued function, and project back.
result We provide error bounds for the approximation of manifold-valued function derivatives.
The paper introduces new functionals and equations for complex vector bundles.
problem Complex vector bundle equations and their solutions.
method Introducing generalized Donaldson's functionals and discussing their Euler-Lagrange equations.
result Uniqueness of solutions to complex vector bundle equations.
The paper characterizes Riemannian manifolds using concircular vector fields and a connecting function.
problem Characterizing Riemannian manifolds using concircular vector fields.
method Introducing a connecting function that links concircular vector fields to potential functions.
result The connecting function is crucial for characterizing n-sphere and Euclidean space. This paper explores geometric and analytic aspects of Lojasiewicz inequalities on vector bundles.
problem Understanding growth and stability conditions for real-analytic functions over vector bundles.
method Outline theory of functionals and variational problems over vector bundles, explore applications to real-analytic functionals.
result Describes the energy functional on $S^{n-1$ as a functional over a vector bundle.
New bound on Rademacher complexity for vector functions.
problem Bounding Rademacher complexity for vector-valued functions.
method Bounding Rademacher complexity by coordinate-wise complexity with a factor of sqrt(K).
result Rademacher complexity is bounded by the maximum coordinate-wise complexity times sqrt(K).
A function that optimally aligns a timelike vector field with its gradients
problem Finding a time function that aligns a timelike vector field with its gradients
method Introducing a functional that penalizes null gradients and minimizes misalignment
result Proving the existence of a unique alignment time function under suitable conditions
We prove implicit function theorems for mappings on topological vector spaces over valued fields. In the real and complex cases, we obtain implicit function theorems for mappings from arbitrary (not necessarily locally convex) topological vector spaces to Banach spaces.
The paper explores MAE as a loss function for DNN vector-to-vector regression, proving its advantages over MSE.
problem Improving loss function for deep neural network based vector-to-vector regression.
method Presenting performance bounds and new properties of MAE, deriving generalized upper bounds, and interpreting MAE as a Laplacian distribution.
result MAE is a more suitable loss function than MSE for DNN based vector-to-vector regression, especially when errors follow a Laplacian distribution.
Two new methods improve forecasting of functional time series data.
problem Forecasting of functional time-dependent data.
method Functional Singular Spectrum Analysis (FSFA) based forecasting methods.
result Our methods outperform existing algorithms for periodic stochastic processes.
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…
Gradient boosting adapted for vector inputs.
problem Applying gradient boosting to multi-class classification problems.
method Extended gradient boosting framework to vector inputs using histogram-based decision trees.
result Efficient algorithm for vector-valued objectives.
The paper proves the existence of a complete holomorphic vector field on a complex manifold with a Kähler-Einstein metric.
problem Existence of complete holomorphic vector fields on complex manifolds with specific metrics.
method Method of potential scaling to find a potential function with constant length differential, then constructing a vector field from its gradient.
result A complete holomorphic vector field is constructed on a complex manifold with a Kähler-Einstein metric.
Paper studies gradient fields from discrete Morse functions for watershed-cut computation.
problem Computing watershed-cuts from discrete Morse functions.
method Discrete Morse Theory and simplicial stacks.
result Minimum Spanning Forest of dual graph is induced by gradient vector field.
A celebrated theorem of Hadwiger states that the Euler-Poincaré characteristic is the the unique invariant and continuous valuation on the distributive lattice of compact polyhedra in R^n that assigns value one to each convex non-empty such polyhedron. This paper provides an analogue of Hadwiger's result for finitely p…
Characterizes magnetic unit vector fields on Lie groups.
problem Classifying magnetic unit vector fields on Lie groups.
method Characterization through critical points of Landau Hall and Dirichlet energy functionals.
result Classification of all magnetic left invariant unit vector fields on 3-dimensional Lie groups.
The paper proposes methods to find a shared active subspace for multivariate vector-valued functions.
problem Minimizing the deviation between function evaluations in the original and reconstructed spaces.
method Manipulating gradients or SPD matrices to identify a shared structure.
result Summing SPD matrices often identifies the best shared active subspace.
Proves properties of Morse vector fields on compact manifolds.
problem Properties of gradient vector fields of Morse functions.
method Analyzes connectedness of critical points and shrinkage of flow.
result Shows connectedness of critical points through orbits and exponential shrinkage.
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 …
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.
We provide some examples of harmonic unit vector fields as normalized gradients of isoparametric functions from a K-contact geometry setting.
A new definition for vector fields extends the Jacobi set concept.
problem Describing interactions between vector fields on complex domains.
method Piecewise linear approach for simplicial complexes.
result Generalizes Jacobi set concept to vector fields.
We present a framework to derive risk bounds for vector-valued learning with a broad class of feature maps and loss functions. Multi-task learning and one-vs-all multi-category learning are treated as examples. We discuss in detail vector-valued functions with one hidden layer, and demonstrate that the conditions under…
Proposes an L1-regularized functional SVM for binary classification with functional covariates.
problem Binary classification with multivariate functional covariates.
method L1-regularized functional support vector machine (SVM) with an accompanying algorithm.
result The proposed classifier performs well in prediction and feature selection.
The study identifies all possible vector field structures on specific 2D shapes.
problem Optimal discrete gradient vector fields on surfaces with 1-2 critical cells.
method Analysis of discrete vector fields on 2D shapes with minimal critical cells.
result All possible structures of discrete Morse functions on specified shapes.
Problems in machine learning (ML) can involve noisy input data, and ML classification methods have reached limiting accuracies when based on standard ML data sets consisting of feature vectors and their classes. Greater accuracy will require incorporation of prior structural information on data into learning. We study …
Support vector machines have attracted much attention in theoretical and in applied statistics. Main topics of recent interest are consistency, learning rates and robustness. In this article, it is shown that support vector machines are qualitatively robust. Since support vector machines can be represented by a functio…
Proves a Gel'fand-Kolmogoroff type result for vector bundles and polynomial functions.
problem Characterizing vector bundles and polynomial functions using differential operators.
method Analyzes the associative structure of symbols of differential operators and their eigenvectors.
result Derives a Gel'fand-Kolmogoroff type result for the algebra of symbols of differential operators.
Predict covariance from features using convex optimization.
problem Predicting the covariance of a Gaussian vector from another feature vector.
method A generalized linear model with convex optimization for fitting parameters.
result Predicted covariance matrices are symmetric positive definite.
The paper extends consistency results for sequential design strategies to vector-valued Gaussian processes.
problem Estimating excursion sets of vector-valued Gaussian processes.
method Clarifying the connection between continuous Gaussian processes and Gaussian measures in Banach spaces, extending concepts and properties from scalar-valued settings to vector-valued settings.
result Consistency results for sequential design strategies can be applied to vector-valued Gaussian processes.
TVS-FNNs can approximate any continuous function on expanded input spaces.
problem Processing a broader range of inputs like sequences and matrices.
method Proving a universal approximation theorem for TVS-FNNs.
result TVS-FNNs can approximate any continuous function on expanded input spaces.
Study vector fields and derivations on differentiable stacks.
problem Understanding structures on differentiable stacks.
method Introduced module structures on dgla of multiplicative vector fields and graded algebra of functions on Lie groupoids.
result Associated structure of a graded Lie-Rinehart algebra on vector fields is Morita invariant.
Analytic torsion equals dynamical zeta function for certain bundles.
problem Equalities between analytic torsion and dynamical zeta functions.
method Analytic torsion and Ruelle dynamical zeta function for admissible twists.
result Generalization of previous results to admissible twists.
We introduce a new functional measure of tail dependence for weakly dependent (asymptotically independent) random vectors, termed weak tail dependence function. The new measure is defined at the level of copulas and we compute it for several copula families such as the Gaussian copula, copulas of a class of Gaussian mi…
Complex valued analytic torsion and dynamical zeta function studied on locally symmetric spaces.
problem Analyzing the Ruelle dynamical zeta function on locally symmetric spaces with flat vector bundles.
method Meromorphic extension and regularisation of the dynamical zeta function, relating it to the complex valued analytic torsion.
result The leading term of the dynamical zeta function at zero is related to the regularised determinant of the flat Laplacian.
Universal approximation for ODENet and ResNet with a single activation function.
problem Approximating complex dynamical systems with limited vector fields.
method Examined ODENet and ResNet with vector fields composed of a single activation function and affine mapping.
result ODENet and ResNet with restricted vector fields can uniformly approximate those with general vector fields.
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…
Learning a distance function or metric on a given data manifold is of great importance in machine learning and pattern recognition. Many of the previous works first embed the manifold to Euclidean space and then learn the distance function. However, such a scheme might not faithfully preserve the distance function if t…
The support vector machine (SVM) is an important class of learning machines for function approach, pattern recognition, and time-serious prediction, etc. It maps samples into the feature space by so-called support vectors of selected samples, and then feature vectors are separated by maximum margin hyperplane. The pres…
The paper uses MDM theory to analyze multifiltering functions on simplicial complexes.
problem Understanding multifiltering functions through discrete Morse theory.
method Applying multiparameter discrete Morse theory to vector-valued multifiltering functions.
result Any multifiltering function can be approximated by a compatible MDM function.
In this article a relation between curvature functionals for surfaces in the Euclidean space and area functionals in relative differential geometry will be given. Relative differential geometry can be described as the geometry of surfaces in the affine space, endowed with a distinguished "relative normal vector field" …
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. Hermitian-Einstein metrics linked to stability of bundles on orbifolds.
problem Existence of Hermitian-Einstein metrics on stable vector bundles over compact Kähler orbifolds.
method Equivalence of slope stability to the existence of Hermitian-Einstein metrics and properness of a functional.
result Equivalence of Hermitian-Einstein metrics and slope stability for stable vector bundles.
Let M be a smooth (C∞) manifold, F1,...,Fn be vector fields on M generating the corresponding flows Φ1,...,Φn, and α1,...,αn:M→R smooth functions. Define the following map f:M→M by f(x)=Φn(...(Φ2(Φ1(x,α1(x)),α2(x)),...,αn(x)). In this note we give a necessa…
A mathematical paradox shows secant planes don't always form a tangent plane, but some analogies hold with a specific vector product.
problem Secant planes of a two-variable smooth function do not always form a tangent plane, even for simple polynomials.
method Analogies with the one-variable case are explored, using Clifford's geometric vector product.
result Some analogies with the one-variable case still hold in the multi-variable context with a specific vector product.
Study vector fields and flows on singular spaces like submanifolds.
problem Understanding vector fields and flows on singular spaces.
method Integrate derivations of the C∞-ring of global smooth functions into flows. result Derivations integrate to smooth flows on subcartesian spaces.
Study Kähler geometry on vector bundles over elliptic curves.
problem Characterize Kähler metrics on vector bundle total spaces.
method Analyzing function theory and Kähler geometry on vector bundles of degree zero.
result Biholomorphic total spaces correspond to isomorphic vector bundles.
A new distortion measure optimizes function approximations in vector quantization.
problem Measuring the quality of vector quantization points for natural signals.
method A canonical distortion measure (CDM) is introduced, induced by an environment of functions on input space.
result Optimizing reconstruction error with respect to CDM yields optimal piecewise constant approximations.