Related papers: Kernel Methods on Riemannian Manifolds with Gaussi…
We develop a framework for function classes generated by parametric ridge kernels: one-dimensional kernels composed with affine projections and averaged over a parameter measure. The induced kernels are positive definite, and the resulting…
Graph convolutional networks (GCNs) are powerful frameworks for learning embeddings of graph-structured data. GCNs are traditionally studied through the lens of Euclidean geometry. Recent works find that non-Euclidean Riemannian manifolds…
The affine Grassmannian is a noncompact smooth manifold that parameterizes all affine subspaces of a fixed dimension. It is a natural generalization of Euclidean space, points being zero-dimensional affine subspaces. We will realize the…
Kernel methods approximate nonlinear maps in a data-driven manner by projecting the target map onto a finite-dimensional Hilbert space called the solution space. Traditionally, this space is a subspace of a fixed ambient reproducing kernel…
Deep neural networks have become the main work horse for many tasks involving learning from data in a variety of applications in Science and Engineering. Traditionally, the input to these networks lie in a vector space and the operations…
Amidst the growing interest in nonparametric regression, we address a significant challenge in Gaussian processes(GP) applied to manifold-based predictors. Existing methods primarily focus on low dimensional constrained domains for heat…
Providing a model that achieves a strong predictive performance and is simultaneously interpretable by humans is one of the most difficult challenges in machine learning research due to the conflicting nature of these two objectives. To…
Choosing the most adequate kernel is crucial in many Machine Learning applications. Gaussian Process is a state-of-the-art technique for regression and classification that heavily relies on a kernel function. However, in the Gaussian…
In this work we develop a novel and foundational framework for analyzing general Riemannian functional data, in particular a new development of tensor Hilbert spaces along curves on a manifold. Such spaces enable us to derive Karhunen-Loeve…
This work aims to prove that the classical Gaussian kernel, when defined on a non-Euclidean symmetric space, is never positive-definite for any choice of parameter. To achieve this goal, the paper develops new geometric and analytical…
Kernel-based classification methods, particularly the support vector machine (SVM), are among the most common algorithms for hyperspectral data classification. The Radial Basis function (RBF) kernel has earned great popularity in…
Kernel methods in machine learning use a kernel function that takes two data points as input and returns their inner product after mapping them to a Hilbert space, implicitly and without actually computing the mapping. For many kernel…
A reproducing kernel can define an embedding of a data point into an infinite dimensional reproducing kernel Hilbert space (RKHS). The norm in this space describes a distance, which we call the kernel distance. The random Fourier features…
We introduce in this paper the mechanism of graph random features (GRFs). GRFs can be used to construct unbiased randomized estimators of several important kernels defined on graphs' nodes, in particular the regularized Laplacian kernel. As…
Computing a consensus object from a set of given objects is a core problem in machine learning and pattern recognition. One popular approach is to formulate it as an optimization problem using the generalized median. Previous methods like…
We propose practical deep Gaussian process models on Riemannian manifolds, similar in spirit to residual neural networks. With manifold-to-manifold hidden layers and an arbitrary last layer, they can model manifold- and scalar-valued…
The random Fourier features (RFFs) method is a powerful and popular technique in kernel approximation for scalability of kernel methods. The theoretical foundation of RFFs is based on the Bochner theorem that relates symmetric, positive…
The Grassmannian manifold G(k, n) serves as a fundamental tool in signal processing, computer vision, and machine learning, where problems often involve classifying, clustering, or comparing subspaces. In this work, we propose a…
We use methods from the Fock space and Segal-Bargmann theories to prove several results on the Gaussian RBF kernel in complex analysis. The latter is one of the most used kernels in modern machine learning kernel methods, and in support…
With view to applications in stochastic analysis and geometry, we introduce a new correspondence for positive definite kernels (p.d.) $K$ and their associated reproducing kernel Hilbert spaces. With this we establish two kinds of…