Related papers: Sampling discretization of integral norms
We develop an asymptotic theory for $L^2$ norms of sample mean vectors of high-dimensional data. An invariance principle for the $L^2$ norms is derived under conditions that involve a delicate interplay between the dimension $p$, the sample…
By using Liu's $q$-partial differential equations theory, we prove that if an analytic function in several variables satisfies a system of $q$-partial differential equations, if and only if it can be expanded in terms of homogeneous…
In many applications sampled data are collected in irregular fashion or are partly lost or unavailable. In these cases it is required to convert irregularly sampled signals to regularly sampled ones or to restore missing data. In this…
A novel approach is proposed to establish a sharp upper bound on the expected supremum of a separable martingale random field, serving as an alternative to classical universal chaining-based methods. The proposed approach begins by deriving…
Functions of interest are often smooth and sparse in some sense, and both priors should be taken into account when interpolating sampled data. Classical linear interpolation methods are effective under strong regularity assumptions, but…
Anomalous diffusion and power-law distributions are observed in various complex systems. To provide a consistent dynamical foundation for these phenomena, we present a geometric derivation of the nonlinear Fokker-Planck equation by…
We prove a generalisation of the disintegration theorem to the setting of multifunctions between Polish probability spaces. Whereas the classical disintegration theorem guarantees the disintegration of a probability measure along the…
We consider the problem of testing mutual independence among the components of a high-dimensional random vector. Building on the rank-based max-sum framework, we introduce fixed finite-$L_q$ power-sum statistics under three general classes…
For control systems in discrete time, this paper discusses measure-theoretic invariance entropy for a subset Q of the state space with respect to a quasi-stationary measure obtained by endowing the control range with a probability measure.…
We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=f \quad\quad\text{in}\quad\quad…
Finding a sparse representation of a possibly noisy signal can be modeled as a variational minimization with l_q-sparsity constraints for q less than one. Especially for real-time, on-line, or iterative applications, in which problems of…
The total generalized variation extends the total variation by incorporating higher-order smoothness. Thus, it can also suffer from similar discretization issues related to isotropy. Inspired by the success of novel discretization schemes…
For an arbitrary given span of high-dimensional multivariate Chebyshev polynomials, an approach to construct spatial discretizations is presented, i.e., the construction of a sampling set that allows for the unique reconstruction of each…
We adapt the theory of normal and special polynomials from symbolic integration to the summation setting, and then built up a general framework embracing both the usual shift case and the $q$-shift case. In the context of this general…
A procedure for obtaining a "minimal" discretization of a partial differential equation, preserving all of its Lie point symmetries is presented. "Minimal" in this case means that the differential equation is replaced by a partial…
We revisit the numerical calculation of generalized Lyapunov exponents, $L$($q$), in deterministic dynamical systems. The standard method consists of adding noise to the dynamics in order to use importance sampling algorithms. Then $L$($q$)…
We propose a framework for computing, optimizing and integrating with respect to a smooth marginal likelihood in statistical models that involve high-dimensional parameters/latent variables and continuous low-dimensional hyperparameters.…
Successive differences on a sequence of data help to discover some smoothness features of this data. This was one of the main reasons for rewriting the classical interpolation formula in terms of such data differences. The aim of this paper…
This paper extends previous work on finitedifference schemes over staggered grids for infinite-dimensional port-Hamiltonian systems. In the one-dimensional setting, it generalizes the discretization approach originally developed for the…
We consider linear inverse problems under white noise. These types of problems can be tackled with, e.g., iterative regularisation methods and the main challenge is to determine a suitable stopping index for the iteration. Convergence…