arXiv subjects

Kateryna Pozharska

Publications and source records attributed to Kateryna Pozharska.

7 recordsLinked to original sources

arXiv2026

Greedy sampling designs via reduced basis methods: optimal recovery in the uniform norm

We study optimal sampling recovery in reproducing kernel Hilbert spaces (RKHS) in the uniform norm. For every RKHS with bounded kernel, we establish new comparisons between linear sampling widths and Gelfand widths that overcome the known square-root gap, without requiring a measure or a Christoffel-type condition. Our bounds rely on nested sampling designs obtained by kernel interpolation at (weak) P-greedy points. Under additional (polynomial) decay assumptions the decay rate of the Gelfand widths directly transfers to the sampling widths. With either a logarithmic oversampling or passing to the square root of the Gelfand widths we obtain a direct comparison (requiring no decay assumption) between them. This is particularly effective for super-polynomial decay, such as in Paley-Wiener spaces. Our results follow from representations of both widths in terms of kernel translates and yield, in the opposite direction, a new existence result for a sharp reduced basis selection. Numerical experiments for Legendre, mixed-Sobolev, and Paley-Wiener kernels illustrate our findings.

math.NA↗

arXiv2026

High-dimensional sparse recovery from function samples Decoders, guarantees and instance optimality

We investigate the reconstruction of multivariate functions from samples using sparse recovery techniques. For Square Root Lasso, Orthogonal Matching Pursuit, and Compressive Sampling Matching Pursuit, we demonstrate both theoretically and empirically that they allow us to recover functions from a small number of random samples. In contrast to Basis Pursuit Denoising, the deployed decoders only require a search space $V_J$ spanned by dictionary elements indexed by $J$ and a sparsity parameter $n$ to guarantee an $L_2$-approximation error decaying no worse than a best $n$-term approximation error and the truncation error with respect to the search space $V_J$ and the uniform norm. We show that this happens simultaneously for all admissible functions if the number of samples scales as $n\log^2 n\log |J|$, coming from known bounds for the RIP for matrices built upon bounded orthonormal systems. As a consequence, we obtain bounds for sampling widths in function classes. In addition, we establish lower bounds on the required sample complexity, which show that the log-factor in $\vert J \vert$ is indeed necessary to obtain such {\em instance-optimal} error guarantees. Finally, we conduct several numerical experiments to show that our theoretical bounds are reasonable and compare the discussed decoders in practice.

math.NA↗

arXiv2025

Sampling recovery in $L_2$ and other norms

We study the recovery of functions in various norms, including $L_p$ with $1\le p\le\infty$, based on function evaluations. We obtain worst case error bounds for general classes of functions in terms of the best $L_2$-approximation from a given nested sequence of subspaces and the Christoffel function of these subspaces. In the case $p=\infty$, our results imply that linear sampling algorithms are optimal up to a constant factor for many reproducing kernel Hilbert spaces.

math.NA↗

arXiv2025

Sampling projections in the uniform norm

We show that there are sampling projections on arbitrary $n$-dimensional subspaces of $B(D)$ with at most $2n$ samples and norm of order $\sqrt{n}$, where $B(D)$ is the space of complex-valued bounded functions on a set $D$. This gives a more explicit form of the Kadets-Snobar theorem for the uniform norm and improves upon Auerbach's lemma. We discuss consequences for optimal recovery in $L_p$.

math.FA↗

arXiv2024

Exact discretization, tight frames and recovery via D-optimal designs

$D$-optimal designs originate in statistics literature as an approach for optimal experimental designs. In numerical analysis points and weights resulting from maximal determinants turned out to be useful for quadrature and interpolation. Also recently, two of the present authors and coauthors investigated a connection to the discretization problem for the uniform norm. Here we use this approach of maximizing the determinant of a certain Gramian matrix with respect to points and weights for the construction of tight frames and exact Marcinkiewicz-Zygmund inequalities in $L_2$. We present a direct and constructive approach resulting in a discrete measure with at most $N \leq n^2+1$ atoms, which discretely and accurately subsamples the $L_2$-norm of complex-valued functions contained in a given $n$-dimensional subspace. This approach can as well be used for the reconstruction of functions from general RKHS in $L_2$ where one only has access to the most important eigenfunctions. We verifiably and deterministically construct points and weights for a weighted least squares recovery procedure and pay in the rate of convergence compared to earlier optimal, however probabilistic approaches. The general results apply to the $d$-sphere or multivariate trigonometric polynomials on $\mathbb{T}^d$ spectrally supported on arbitrary finite index sets~$I \subset \mathbb{Z}^d$. They can be discretized using at most $|I|^2-|I|+1$ points and weights. Numerical experiments indicate the sharpness of this result. As a negative result we prove that, in general, it is not possible to control the number of points in a reconstructing lattice rule only in the cardinality $|I|$ without additional condition on the structure of $I$. We support our findings with numerical experiments.

math.NA↗

arXiv2024

The best $m$-term trigonometric approximations of the classes of periodic functions of one and many variables in the space $B_{q,1}$

Exact order estimates are obtained of the best $m$-term trigonometric approximations of the Nikol'skii-Besov classes $B^r_{p, θ}$ of periodic functions of one and many variables in the space $B_{q,1}$. In the univariate case ($d=1$), we get the orders of the respective approximation characteristics on the classes $B^r_{p, θ}$ as well as on the Sobolev classes $W^r_{p, {\boldsymbolα}}$ in the space $B_{\infty,1}$ in the case $1\leq p \leq \infty$.

math.CA↗

arXiv2021

A note on sampling recovery of multivariate functions in the uniform norm

We study the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. Our main interest is to obtain preasymptotic estimates for the corresponding sampling numbers. We obtain results in terms of the decay of related singular numbers of the compact embedding into $L_2(D,\varrho_D)$ multiplied with the supremum of the Christoffel function of the subspace spanned by the first $m$ singular functions. Here the measure $\varrho_D$ is at our disposal. As an application we obtain near optimal upper bounds for the sampling numbers for periodic Sobolev type spaces with general smoothness weight. Those can be bounded in terms of the corresponding benchmark approximation number in the uniform norm, which allows for preasymptotic bounds. By applying a recently introduced sub-sampling technique related to Weaver's conjecture we mostly lose a $\sqrt{\log n}$ and sometimes even less. Finally we point out a relation to the corresponding Kolmogorov numbers.

math.NA↗