Related papers: Maximum mean discrepancies of Farey sequences

We propose a novel combination of optimization tools with learning theory bounds in order to analyze the sample complexity of optimal kernel sum classifiers. This contrasts the typical learning theoretic results which hold for all…

The merit factor of a $\{-1, 1\}$ binary sequence measures the collective smallness of its non-trivial aperiodic autocorrelations. Binary sequences with large merit factor are important in digital communications because they allow the…

This note responds to "Promises and Pitfalls of Deep Kernel Learning" (Ober et al., 2021). The marginal likelihood of a Gaussian process can be compartmentalized into a data fit term and a complexity penalty. Ober et al. (2021) shows that…

Defining suitable quantum extensions of classical divergences often poses a challenge due to the non-commutative nature of quantum information. In this work, we propose a new approach via what we call the layer cake representation. The…

We consider piecewise linear interpolation from the perspective of kernel interpolation and quadrature. If the Sobolev space $W_2^1(0, 1)$ is equipped with a suitable inner product, its reproducing kernel is piecewise linear and gives rise…

We study fast algorithms for computing fundamental properties of a positive semidefinite kernel matrix $K \in \mathbb{R}^{n \times n}$ corresponding to $n$ points $x_1,\ldots,x_n \in \mathbb{R}^d$. In particular, we consider estimating the…

We introduce a divergence measure between data distributions based on operators in reproducing kernel Hilbert spaces defined by kernels. The empirical estimator of the divergence is computed using the eigenvalues of positive definite Gram…

We study kernel methods in machine learning from the perspective of feature subspace. We establish a one-to-one correspondence between feature subspaces and kernels and propose an information-theoretic measure for kernels. In particular, we…

Through an equivalent condition on the Farey series set forth by Franel and Landau, we prove Riemann Hypothesis for the Riemann zeta-function and the Dirichlet L-function.

We prove that conditional diffusion models whose reverse kernels are finite Gaussian mixtures with ReLU-network logits can approximate suitably regular target distributions arbitrarily well in context-averaged conditional KL divergence, up…

Marginalising over families of Gaussian Process kernels produces flexible model classes with well-calibrated uncertainty estimates. Existing approaches require likelihood evaluations of many kernels, rendering them prohibitively expensive…

Kernel matrices are a key quantity in kernel-based approximation, and important properties such as stability and algorithmic convergence can be analyzed with their help. In this work we refine a multivariate Ingham-type theorem, which is…

The optimized quantum $f$-divergences form a family of distinguishability measures that includes the quantum relative entropy and the sandwiched R\'enyi relative quasi-entropy as special cases. In this paper, we establish physically…

Mean embeddings provide an extremely flexible and powerful tool in machine learning and statistics to represent probability distributions and define a semi-metric (MMD, maximum mean discrepancy; also called N-distance or energy distance),…

We consider a large class of piecewise expanding maps T of [0,1] with a neutral fixed point, and their associated Markov chain Y_i whose transition kernel is the Perron-Frobenius operator of T with respect to the absolutely continuous…

We prove that the Farey sequences can be express into equivalence classes labeled by a fractal parameter which looks like a Hausdorff dimension $h$ defined within the interval 1 < h < 2. The classes $h$ satisfy the same properties of the…

Given any generically \'etale morphism of varieties $f \colon X \to Y$, we define the relative Mather discrepancy function on the arc space $X_\infty$ of the domain and show that this function computes the dimension of the kernel of the…

Analytical expressions are derived for the number of fractions with equal numerators in the Farey sequence of order $n$, $F_n$, and in the truncated Farey sequence $F_n^{1/k}$ containing all Farey fractions below $1/k$, with $1\leq k \leq…

We consider the problem of metric learning subject to a set of constraints on relative-distance comparisons between the data items. Such constraints are meant to reflect side-information that is not expressed directly in the feature vectors…

Kernel mean embeddings are a popular tool that consists in representing probability measures by their infinite-dimensional mean embeddings in a reproducing kernel Hilbert space. When the kernel is characteristic, mean embeddings can be used…