arXiv subjects

Konstantin Ryutin

Publications and source records attributed to Konstantin Ryutin.

4 recordsLinked to original sources

arXiv2025

Kolmogorov widths of balls in mixed norms: the case of rigidity

We describe the set of parameters $(p_1,p_2,q_1,q_2)$ such that the balls $B_{q_1,q_2}^{s,b}$ are rigid in $\ell_{q_1,q_2}^{s,b}$ metric i.e. they are poorly approximated by linear subspaces of dimension $\le (1-\varepsilon)sb$, for large $s, b$. Thus we have settled an important qualitative case in the problem of estimating widths of balls in mixed norms. The proof combines lower bounds from our previous papers and a new construction for the approximation by linear subspaces in the so-called exceptional case.

math.FA↗

arXiv2024

Widths and rigidity of unconditional sets and random vectors

We prove that any unconditional set in $\mathbb{R}^N$ that is invariant under cyclic shifts of coordinates is rigid in $\ell_q^N$, $1\le q\le 2$, i.e. it can not be well approximated by linear spaces of dimension essentially smaller than $N$. We apply the approach of E.D.~Gluskin to the setting of averaged Kolmogorov widths of unconditional random vectors or vectors of independent mean zero random variables, and prove their rigidity. These results are obtained using a general lower bound for the averaged Kolmogorov width via weak moments of biorthogonal random vector. This paper continues the study of the rigidity initiated by the first author. We also provide several corollaries including lower bounds for Kolmogorov widths of mixed norm balls.

math.FA↗

arXiv2023

Polynomial approximation on disjoint segments and amplification of approximation

We construct explicit easily implementable polynomial approximations of sufficiently high accuracy for locally constant functions on the union of disjoint segments. This problem has important applications in several areas of numerical analysis, complexity theory, quantum algorithms, etc. The one, most relevant for us, is the amplification of approximation method: it allows to construct approximations of higher degree $M$ and better accuracy from the approximations of degree $m$.

math.FA↗

arXiv2021

Recovery of regular ridge functions on the ball

We consider the problem of the uniform (in $L_\infty$) recovery of ridge functions $f(x)=φ(\langle a,x\rangle)$, $x\in B_2^n$, using noisy evaluations $y_1\approx f(x^1),\ldots,y_N\approx f(x^N)$. It is known that for classes of functions $φ$ of finite smoothness the problem suffers from the curse of dimensionality: in order to provide good accuracy for the recovery it is necessary to make exponential number of evaluations. We prove that if $φ$ is analytic in a neighborhood of $[-1,1]$ and the noise is very small, $\varepsilon\le\exp(-c\log^2n)$, then there is an efficient algorithm that recovers $f$ with good accuracy using $O(n\log^2n)$ function evaluations.

math.FA↗