arXiv research

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

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

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…

Optimal rates for vector-valued regression on various norms.

Abstract: Generalizes multisymplectic forms to vector-valued versions.

The classical Schläfli formula, and its ``higher'' analogs given in [SS03], are relations between the variations of the volumes and ``curvatures'' of faces of different dimensions of a polyhedra (which can be Euclidean, spherical or hyperbolic) under a first-order deformation. We describe here analogs of those formulas…

Study efficient neural operator learning using variation spaces.

The space of vector-valued forms on any manifold is a graded Lie algebra with respect to the Frolicher-Nijenhuis bracket. In this paper we consider multiplicative vector-valued forms on Lie groupoids and show that they naturally form a graded Lie subalgebra. Along the way, we discuss various examples and different char…

Some intrinsic tools from the formal theory of variational equations are being demonstrated at work in application to one concrete example of the third-order evolution equation of free relativistic top in three-dimensional space-time. The main goal is to introduce a combined approach consisting in the simultaneous util…

Develops vector-valued RKBS for neural networks and operators.

The paper develops a new approach to solve vector-valued PDEs on manifolds with minimal regularity.

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

We examine the relationship between nonabelian Hodge theory for Riemann surfaces and the theory of vector valued modular forms. In particular, we explain how one might use this relationship to prove a conjectural three-term inequality on the weights of free bases of vector valued modular forms associated to complex, fi…

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

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

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

New method transfers emotions in facial images.

Vector-valued learning, where the output space admits a vector-valued structure, is an important problem that covers a broad family of important domains, e.g. multi-task learning and transfer learning. Using local Rademacher complexity and unlabeled data, we derive novel semi-supervised excess risk bounds for general v…

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

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

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

tvGP-VAE models tensor-valued latent variables with Gaussian processes for better data structure representation.

In this study, we propose a new definition of multivariate conditional value-at-risk (MCVaR) as a set of vectors for discrete probability spaces. We explore the properties of the vector-valued MCVaR (VMCVaR) and show the advantages of VMCVaR over the existing definitions given for continuous random variables when adapt…

As in a symmetric space of noncompact type, one can associate to an oriented geodesic segment in a Euclidean building a vector valued length in the Euclidean Weyl chamber; in addition to the metric length it contains information on the direction of the segment. We study in this paper restrictions on the vector valued s…

Gaussian processes adapted for Riemannian manifolds using gauge-independent kernels.

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

In [16], a new family of vector-valued risk measures called multivariate expectiles is introduced. In this paper, we focus on the asymptotic behavior of these measures in a multivariate regular variations context. For models with equivalent tails, we propose an estimator of these multivariate asymptotic expectiles, in …

Framework for transferring discount curve estimates across fixed-income product classes.

We identify the Variational Principle governing inifinity-Harmonic maps, that is solutions to the Infinity-Laplacian. The system was first derived in the limit of the p-Laplacian as p->inifinity in [K2] and is recently studied in [K3]. Here we show that it is the "Euler-Lagrange PDE" of vector-valued Calculus of Variat…

We prove the existence of a continuous $BV$ minimizer with $C^{0}$ boundary value for the $p$-area (pseudohermitian or horizontal area) in a parabolically convex bounded domain. We extend the domain of the area functional from $BV$ functions to vector-valued measures. Our main purpose is to study the first and second v…

The report analyzes infinite-dimensional output space regression.

We discuss sharp Sobolev inequalities for vector valued maps.

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…

Boosting framework for vector-valued prediction with geometric stability.

Study optimizes online learning for vector-valued data regression.

We present a novel extension of multi-output Gaussian processes for handling heterogeneous outputs. We assume that each output has its own likelihood function and use a vector-valued Gaussian process prior to jointly model the parameters in all likelihoods as latent functions. Our multi-output Gaussian process uses a c…

In this article we introduce a diffeomorphism-invariant Riemannian metric on the space of vector valued one-forms. The particular choice of metric is motivated by potential future applications in the field of functional data and shape analysis and by connections to the Ebin metric on the space of all Riemannian metrics…

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

We prove realizability theorems for vector-valued polynomial mappings, real-algebraic sets and compact smooth manifolds by moduli spaces of planar linkages. We also establish a relation between universality theorems for moduli spaces of mechanical linkages and projective arrangements.

This paper presents a methodology and numerical algorithms for constructing accelerated gradient flows on the space of probability distributions. In particular, we extend the recent variational formulation of accelerated gradient methods in (wibisono, et. al. 2016) from vector valued variables to probability distributi…

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…

We consider the problem of metric learning for multi-view data and present a novel method for learning within-view as well as between-view metrics in vector-valued kernel spaces, as a way to capture multi-modal structure of the data. We formulate two convex optimization problems to jointly learn the metric and the clas…

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

Proposes Siegel neural networks for improved classification tasks.

This paper presents a generalization of symplectic geometry to a principal bundle over the configuration space of a classical field. This bundle, the vertically adapted linear frame bundle, is obtained by breaking the symmetry of the full linear frame bundle of the field configuration space, and it inherits a generaliz…

In computer vision, image datasets used for classification are naturally associated with multiple labels and comprised of multiple views, because each image may contain several objects (e.g. pedestrian, bicycle and tree) and is properly characterized by multiple visual features (e.g. color, texture and shape). Currentl…

Whitney type examples of maps $f\in C^k(\real^m,\real^n)$ for a maximal possible real $k$, and multidimensional space-filling curves with special properties are constructed.

The paper introduces novel Gaussian process models for vector-valued signals on manifolds.