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Interpolation and approximation of functionals with conditionally positive definite kernels is considered on sets of centers that are not determining for polynomials. It is shown that polynomial consistency is sufficient in order to define…
In this paper we show that the strictly positive definite matrix valued isotropic kernels in the circle and the real dot product kernels in Euclidean spaces are not well behaved with respect to its scalar valued projections. We generalize…
The problem of classification in machine learning has often been approached in terms of function approximation. In this paper, we propose an alternative approach for classification in arbitrary compact metric spaces which, in theory, yields…
In this article we study the field of Hilbertian metrics and positive definit (pd) kernels on probability measures, they have a real interest in kernel methods. Firstly we will make a study based on the Alpha-Beta-divergence to have a…
We provide a characterization for the continuous positive definite kernels on $\mathbb R^d$ that are invariant to linear isometries, i.e. invariant under the orthogonal group $O(d)$. Furthermore, we provide necessary and sufficient…
In spatial statistics and machine learning, the kernel matrix plays a pivotal role in prediction, classification, and maximum likelihood estimation. A thorough examination reveals that for large sample sizes, the kernel matrix becomes…
Similarity plays a fundamental role in many areas, including data mining, machine learning, statistics and various applied domains. Inspired by the success of ensemble methods and the flexibility of trees, we propose to learn a similarity…
Kernel mean embeddings have recently attracted the attention of the machine learning community. They map measures $\mu$ from some set $M$ to functions in a reproducing kernel Hilbert space (RKHS) with kernel $k$. The RKHS distance of two…
Kernel approximation methods create explicit, low-dimensional kernel feature maps to deal with the high computational and memory complexity of standard techniques. This work studies a supervised kernel learning methodology to optimize such…
Approximating non-linear kernels using feature maps has gained a lot of interest in recent years due to applications in reducing training and testing times of SVM classifiers and other kernel based learning algorithms. We extend this line…
The kernel exponential family is a rich class of distributions, which can be fit efficiently and with statistical guarantees by score matching. Being required to choose a priori a simple kernel such as the Gaussian, however, limits its…
The kernel trick concept, formulated as an inner product in a feature space, facilitates powerful extensions to many well-known algorithms. While the kernel matrix involves inner products in the feature space, the sample covariance matrix…
Predictions are often probabilities; e.g., a prediction could be for precipitation tomorrow, but with only a 30% chance. Given such probabilistic predictions together with the actual outcomes, "reliability diagrams" help detect and diagnose…
Many contemporary statistical learning methods assume a Euclidean feature space. This paper presents a method for defining similarity based on hyperspherical geometry and shows that it often improves the performance of support vector…
We consider a kernel based harmonic analysis of "boundary," and boundary representations. Our setting is general: certain classes of positive definite kernels. Our theorems extend (and are motivated by) results and notions from classical…
We study the statistical-computational trade-offs for learning with exact invariances (or symmetries) using kernel regression. Traditional methods, such as data augmentation, group averaging, canonicalization, and frame-averaging, either…
The paper studies strictly positive definite kernels on compact Riemannian manifolds. We state new conditions to ensure strict positive definiteness for general kernels and kernels with certain convolutional structure. We also state…
Exact sequences are a well known notion in homological algebra. We investigate here the more vague properties of 'homotopical exactness', appearing for instance in the fibre or cofibre sequence of a map. Such notions of exactness can be…
Feature maps associated with positive definite kernels play a central role in kernel methods and learning theory, where regularity properties such as Lipschitz continuity are closely related to robustness and stability guarantees. Despite…
For two decades, reproducing kernels and their associated discrepancies have facilitated elegant theoretical analyses in the setting of quasi Monte Carlo. These same tools are now receiving interest in statistics and related fields, as…