arXiv research

Extends optimal regularity and compactness to vector bundles over non-Riemannian manifolds.

We propose an algorithm for exploring the entire regularization path of asymmetric-cost linear support vector machines. Empirical evidence suggests the predictive power of support vector machines depends on the regularization parameters of the training algorithms. The algorithms exploring the entire regularization path…

Study shows limitations of Lie bracket commutation for nonsmooth vector fields.

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 …

In this paper we study the geometrical structures on the cotangent bundle using the notions of adapted tangent structure and regular vector fields. We prove that the dynamical covariant derivative on $T^{*}M$ fix a nonlinear connection for a given $\mathcal{J}$-regular vector field. Using the Legendre transformation in…

Proposes a regularization method for unsupervised domain adaptation that aligns predictions with target data's top singular vectors.

We provide a characterization of r-regular sets in terms of the Lipschitz regularity of normal vector fields to the boundary.

Paper develops reduction theory for controlled Lagrangian systems with symmetry and momentum map.

Study Higgs bundles and flat connections on quasi-regular Sasakian manifolds.

We show that the metrical connection can be introduced in the two-dimensional Finsler space such that entailed parallel transports along curves joining points of the underlying manifold keep the two-vector angle as well as the length of the tangent vector, thereby realizing isometries of tangent spaces under the parall…

Proposes an L1-regularized functional SVM for binary classification with functional covariates.

We show a higher order integrability theorem for distributions generated by a family of vector fields under a horizontal regularity assumption on their coefficients. We use as chart a class of almost exponential maps which we discuss in details

Study on special coordinates for Dubrovin-Frobenius manifolds in low dimensions.

The paper studies implicit regularization in over-parameterized models for high-dimensional data.

We examine the effect of the Group Lasso (gLasso) regularizer in selecting the salient nodes of Deep Neural Network (DNN) hidden layers by applying a DNN-HMM hybrid speech recognizer to TED Talks speech data. We test two types of gLasso regularization, one for outgoing weight vectors and another for incoming weight vec…

Study of generalized Bishop frames on curves in 4D space.

New insights into tail behavior of heavy-tailed random vectors and processes.

A new algorithm for faster model selection in twin multi-class SVM.

We study Tikhonov regularization for solving ill--posed operator equations where the solutions are functions defined on surfaces. One contribution of this paper is an error analysis of Tikhonov regularization which takes into account perturbations of the surfaces, in particular when the surfaces are approximated by spl…

The support vector machine (SVM) is a widely used machine learning tool for classification based on statistical learning theory. Given a set of training data, the SVM finds a hyperplane that separates two different classes of data points by the largest distance. While the standard form of SVM uses L2-norm regularizatio…

Sparse Tucker decomposition with graph regularization improves time series forecasting accuracy.

In representation learning (RL), how to make the learned representations easy to interpret and less overfitted to training data are two important but challenging issues. To address these problems, we study a new type of regulariza- tion approach that encourages the supports of weight vectors in RL models to have small …

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…

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

We look at Poisson geometry taking the viewpoint of singular foliations, understood as suitable submodules generated by Hamiltonian vector fields rather than partitions into (symplectic) leaves. The class of Poisson structures which behave best from this point of view, are those whose submodule generated by Hamiltonian…

Develops regularity theory for Beckmann's optimal transport problem.

Transformer learns context and regularization for ICL in inverse problems.

In this paper, we characterize conformal vector fields of any (regular or singular) $(α,β)$-space with some PDEs. Further, we show some properties of conformal vector fields of a class of singular $(α,β)$-spaces satisfying certain geometric conditions.

For a variety of regularized optimization problems in machine learning, algorithms computing the entire solution path have been developed recently. Most of these methods are quadratic programs that are parameterized by a single parameter, as for example the Support Vector Machine (SVM). Solution path algorithms do not …

Efficient method for choosing data points in machine learning models.

In this paper, we give precisely the geometric constraint conditions of canonical symplectic form and regular reduced symplectic forms for the dynamical vector fields of a regular controlled Hamiltonian (RCH) system and its regular reduced systems, which are called the Type I and Type II of Hamilton-Jacobi equations. A…

Study shows how to balance memory and learning efficiency in continual learning.

In this paper, we study spherical images of the modified orthogonal vector fields and Darboux vector of a regular curve which lies on the unit sphere in Euclidean 3-space.

New concept of regular separation for ODEs leads to improved Hardy field results.

Derives spacetime regularity under specific curvature conditions.

In this paper, we investigate a multivariate multi-response (MVMR) linear regression problem, which contains multiple linear regression models with differently distributed design matrices, and different regression and output vectors. The goal is to recover the support union of all regression vectors using $l_1/l_2$-reg…

Proves helicity is the only regular Casimir for 3D hydrodynamics.

The paper improves the smoothness of vector fields on manifolds.

In nonparametric classification and regression problems, regularized kernel methods, in particular support vector machines, attract much attention in theoretical and in applied statistics. In an abstract sense, regularized kernel methods (simply called SVMs here) can be seen as regularized M-estimators for a parameter …

On a Poisson manifold endowed with a Riemannian metric we will construct a vector field that generalizes the double bracket vector field defined on semi-simple Lie algebras. On a regular symplectic leaf we will construct a generalization of the normal metric such that the above vector field restricted to the symplectic…

We propose in this contribution a method for l one regularization in prototype based relevance learning vector quantization (LVQ) for sparse relevance profiles. Sparse relevance profiles in hyperspectral data analysis fade down those spectral bands which are not necessary for classification. In particular, we consider …

Conditional meta-learning improves meta-learning performance in diverse task environments.

The paper studies regular contact manifolds and their products.

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

We consider a family of $C^1$ vector fields satisfying a suitable higher order involutivity condition. We discuss the definition of commutators, the regularity of Sussmann's orbits and the Poincaré inequality.

Study of generalized Bishop frames on time-like curves in 4D Lorentz space.

In this paper, we propose a novel regularization method, RotationOut, for neural networks. Different from Dropout that handles each neuron/channel independently, RotationOut regards its input layer as an entire vector and introduces regularization by randomly rotating the vector. RotationOut can also be used in convolu…

We propose a new class of convex penalty functions, called \emph{variational Gram functions} (VGFs), that can promote pairwise relations, such as orthogonality, among a set of vectors in a vector space. These functions can serve as regularizers in convex optimization problems arising from hierarchical classification, m…