arXiv research

Abstract: Generalizes multisymplectic forms to vector-valued versions.

This paper studies Lie groupoids and their vector-valued forms.

The paper connects Hodge theory and modular forms to prove inequalities.

Introduces a metric on vector-valued one-forms for functional data analysis.

Completes the space of vector-valued one-forms on manifolds.

Optimal transport for vector Gaussian mixtures improves efficiency and structure preservation.

Study automorphic forms on bounded domains, proving spanning results and estimating norms.

In the first chapter, we give a precise and general description of gerbes valued in arbitrary crossed module and over an arbitrary differential stack. We do it using only Lie groupoids, hence ordinary differential geometry, by considering differential stacks as being Lie groupoids up to Morita equivalence. We prove the…

Study strong max principle for vector maps on curved spaces.

Paper proposes a new method to evaluate joint risk under uncertainty.

New perspective on Cartan geometries using multiplicative forms.

Building on the Utiyama principle we formulate an approach to Lagrangian field theory in which exterior covariant differentials of vector-valued forms replace partial derivatives, in the sense that they take up the role played by the latter in the usual jet bundle formulation. Actually a natural Lagrangian can be writt…

We define covariant Lie derivatives acting on vector-valued forms on Lie algebroids and study their properties. This allows us to obtain a concise formula for the Frölicher-Nijenhuis bracket on Lie algebroids.

The paper extends Hodge-de Rham and Lichnérowicz Laplacians to double forms and proves vanishing theorems.

Let V be the pseudo-Euclidean vector space of signature (p,q), p>2 and W a module over the even Clifford algebra Cl^0 (V). A homogeneous quaternionic manifold (M,Q) is constructed for any spin(V)-equivariant linear map Π: \wedge^2 W \to V. If the skew symmetric vector valued bilinear form Πis nondegenerate then (M,Q) i…

New elastic metrics for surface shape analysis.

In our previous paper (Axiomatic Differential Geometry II-3) we have discussed the general Jacobi identity, from which the Jacobi identity of vector fields follows readily. In this paper we derive Jacobi-like identities of tangent-vector-valued forms from the general Jacobi identity.

Improved bounds and algorithms for vector-valued learning using unlabeled data.

We generalize the prequantization central extension of a group of diffeomorphisms preserving a closed 2-form ω(ω-invariant diffeomorphisms) to an abelian extension of a group of diffeomorphisms preserving a closed vector valued 2-form ω, up to a linear isomorphism (ω-equivariant diffeomorphisms). Every abelian extensio…

The paper extends consistency results for sequential design strategies to vector-valued Gaussian processes.

We discuss sharp Sobolev inequalities for vector valued maps.

Motivated by multi-task machine learning with Banach spaces, we propose the notion of vector-valued reproducing kernel Banach spaces (RKBS). Basic properties of the spaces and the associated reproducing kernels are investigated. We also present feature map constructions and several concrete examples of vector-valued RK…

This paper presents a general vector-valued reproducing kernel Hilbert spaces (RKHS) framework for the problem of learning an unknown functional dependency between a structured input space and a structured output space. Our formulation encompasses both Vector-valued Manifold Regularization and Co-regularized Multi-view…

Optimal rates for vector-valued regression on various norms.

SL(n) covariant valuations on Orlicz spaces are represented and characterized.

Boosting framework for vector-valued prediction with geometric stability.

Study confirms learning rates for vector-valued spectral algorithms, proving consistency.

Study of Killing spinor-valued forms and their integrability conditions.

The paper shows vector-valued risk measures ignore dependence structures.

This study improves graph signal denoising for vector-valued data with non-convex penalties.

The paper proposes methods to find a shared active subspace for multivariate vector-valued functions.

No arbitrage holds if a Pareto solution exists for vector-valued utility maximization.

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…

26 different concrete representations of the space of vector valued distributions on a smooth manifold of dimension n are presented systematically, most of them new. In the particular case of representations as module homomorphisms acting on sections of the dual bundle resp. on n-forms, the continuity of these homomorp…

New definition of MCVaR for discrete probability spaces.

Study improves self-normalized bounds for vector-valued processes beyond sub-Gaussianity.

Study multiplicative tensors on Lie groupoids using infinitesimal data.

We approximate derivatives of functions on manifolds by embedding them and applying vector-valued operators.

Paper introduces vector-valued variation spaces for multi-output neural networks.

Paper analyzes error bounds for learning with vector-valued RF, improving existing analyses.

Randomized algorithm solves vector-valued regression problems with low-rank operators.

Derives Selberg trace formula on Riemann surfaces and generalizes to other spaces.

We demonstrate that it is conceptually and computationally favorable to regard spin-weighted spherical harmonics as vector valued functions on the total space $SO(3)$ of the Hopf bundle, satisfying a covariance condition with respect to the gauge group $U(1)$ of this bundle. A key role is played by the invariant connec…

Novel method for multi-view metric learning in vector-valued kernel spaces.

In this paper we find solutions $u_ε$ to a certain class of vector-valued parabolic Allen-Cahn equation that as $ε\to 0$ develops as interface a given triod evolving under curve shortening flow.

Develops vector-valued RKBS for neural networks and operators.

Paper develops a theory for Patterson-Sullivan measures in higher rank symmetric spaces.

New method transfers emotions in facial images.