Related papers: Error bounds for numerical differentiation using k…
We propose a kernelized classification layer for deep networks. Although conventional deep networks introduce an abundance of nonlinearity for representation (feature) learning, they almost universally use a linear classifier on the learned…
We consider a least-squares variational kernel-based method for numerical solution of second order elliptic partial differential equations on a multi-dimensional domain. In this setting it is not assumed that the differential operator is…
This article presents a three-step framework for learning and solving partial differential equations (PDEs) using kernel methods. Given a training set consisting of pairs of noisy PDE solutions and source/boundary terms on a mesh, kernel…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
The methods commonly used for numerical differentiation, such as the "center-difference formula" and "four-points formula" are unusable in simulations or real-time data analysis because they require knowledge of the future. In Bard'11, an…
Boundary integral equations and Nystrom discretization provide a powerful tool for the solution of Laplace and Helmholtz boundary value problems. However, often a weakly-singular kernel arises, in which case specialized quadratures that…
The so-called tuned-correlated kernel (sometimes also called the first-order stable spline kernel) is one of the most widely used kernels for the regularized impulse response estimation. This kernel can be derived by applying an exponential…
In this paper, we establish an exponential inequality for U-statistics of i.i.d. data, varying kernel and taking values in a separable Hilbert space. The bound are expressed as a sum of an exponential term plus an other one involving the…
Computable and sharp error bounds are derived for asymptotic expansions for linear differential equations having a simple turning point. The expansions involve Airy functions and slowly varying coefficient functions. The sharpness of the…
We estimate on a compact interval densities with isolated irregularities, such as discontinuities or discontinuities in some derivatives. From independent and identically distributed observations we construct a kernel estimator with…
Quantum kernel methods promise enhanced expressivity for learning structured data, but their usefulness has been limited by kernel concentration and barren plateaus. Both effects are mathematically equivalent and suppress trainability. We…
It is well-known that univariate cubic spline interpolation, if carried out on point sets with fill distance $h$, converges only like ${\cal O}(h^2)$ in $L_2[a,b]$ for functions in $W_2^2[a,b]$ if no additional assumptions are made. But…
The classical results about the boundary values of holomorphic or harmonic functions on a domain $D$ state that under additional integrability assumptions these functions have limits along specific sets approaching boundary. The proofs of…
We develop a new class of distance-aware error bounds that tightly characterize the approximation error of spline neural networks. Our bottom-up approach analyzes the error bound of each neuron (a spline) and then extends it to the full…
Application of nonparametric and semiparametric regression techniques to high-dimensional time series data has been hampered due to the lack of effective tools to address the ``curse of dimensionality.'' Under rather weak conditions, we…
The compact fourth-order finite-difference scheme for solving the 1d wave equation is studied. New error bounds of the fractional order $\mathcal{O}(h^{4(\lambda-1)/5})$ are proved in the mesh energy norm in terms of data, for two initial…
Existing uncertainty modeling approaches try to detect an out-of-distribution point from the in-distribution dataset. We extend this argument to detect finer-grained uncertainty that distinguishes between (a). certain points, (b). uncertain…
In this PhD thesis, we propose a novel framework for uncertainty quantification in machine learning, which is based on proper scores. Uncertainty quantification is an important cornerstone for trustworthy and reliable machine learning…
In this paper we provide a priori error estimates in standard Sobolev (semi-)norms for approximation in spline spaces of maximal smoothness on arbitrary grids. The error estimates are expressed in terms of a power of the maximal grid…
Kernel Density Estimation is a very popular technique of approximating a density function from samples. The accuracy is generally well-understood and depends, roughly speaking, on the kernel decay and local smoothness of the true density.…