arXiv subjects

Yuri Malykhin

Publications and source records attributed to Yuri Malykhin.

11 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↗

arXiv2025

Kolmogorov widths of the class $W_1^1$

We prove that $d_n(W^1_1,L_q)\asymp n^{-1/2}\log n$, $2<q<\infty$. This completes the study of orders of decay of Kolmogorov widths for the classical case of the univariate Sobolev classes of integer smoothness.

math.FA↗

arXiv2024

On the structure of low-rank matrices that approximate the identity matrix

Consider a matrix $A$ of rank $n$ that approximates the $N\times N$ identity matrix with elementwise error at most $1/3$. We give a lower bound on the number of elements s.t. $|A_{i,j}|>γ$, for a certain threshold. Two corollaries are obtained. 1. If $n \le K\log N$ with some $K$, then at least $c(K)N^2$ elements satisfy $|A_{i,j}|>c(K)n^{-1/2}$. This answers a question of B.S. Kashin. 2. The number of nonzero elements in $A$ is at least $c\log(N)/(n\log(2+n/\log N))$.

math.FA↗

arXiv2024

One-sided discretization inequalities and sampling recovery

Recently, in a number of papers it was understood that results on sampling discretization and on the universal sampling discretization can be successfully used in the problem of sampling recovery. Moreover, it turns out that it is sufficient to only have a one-sided discretization inequality for some of those applications. This motivates us to write the present paper as a survey/research paper with the focus on the one-sided discretization inequalities and their applications in the sampling recovery. In this sense the paper complements the two existing survey papers on sampling discretization.

math.NA↗

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↗

arXiv2024

Widths and rigidity

We consider Kolmogorov widths of finite sets of functions. Any orthonormal system of $N$ functions is rigid in $L_2$, i.e. it cannot be well approximated by linear subspaces of dimension essentially smaller than $N$. This is not true for weaker metrics: it is known that in every $L_p$, $p<2$, the first $N$ Walsh functions can be $o(1)$-approximated by a linear space of dimension $o(N)$. We give some sufficient conditions for rigidity. We prove that independence of functions (in the probabilistic meaning) implies rigidity in $L_1$ and even in $L_0$ -- the metric that corresponds to convergence in measure. In the case of $L_p$, $1<p<2$, the condition is weaker: any $S_{p'}$-system is $L_p$-rigid. Also we obtain some positive results, e.g. that first $N$ trigonometric functions can be approximated by very-low-dimensional spaces in $L_0$, and by subspaces generated by $o(N)$ harmonics in $L_p$, $p<1$.

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↗

arXiv2022

Matrix and tensor rigidity and $L_p$-approximation

In this paper we apply methods originated in Complexity theory to some problems of Approximation. We notice that the construction of Alman and Williams that disproves the rigidity of Walsh-Hadamard matrices, provides good $\ell_p$-approximation for $p<2$. It follows that the first $n$ functions of Walsh system can be approximated with an error $n^{-δ}$ by a linear space of dimension $n^{1-δ}$: $$ d_{n^{1-δ}}(\{w_1,\ldots,w_n\}, L_p[0,1]) \le n^{-δ},\quad p\in[1,2),\;δ=δ(p)>0. $$ We do not know if this is possible for the trigonometric system. We show that the algebraic method of Alon--Frankl--Rödl for bounding the number of low-signum-rank matrices, works for tensors: almost all signum-tensors have large signum-rank and can't be $\ell_1$-approximated by low-rank tensors. This implies lower bounds for $Θ_m$~ -- the error of $m$-term approximation of multivariate functions by sums of tensor products $u^1(x_1)\cdots u^d(x_d)$. In particular, for the set of trigonometric polynomials with spectrum in $\prod_{j=1}^d[-n_j,n_j]$ and of norm $\|t\|_\infty\le 1$ we have $$ Θ_m(\mathcal T(n_1,\ldots,n_d)_\infty,L_1[-π,π]^d) \ge c_1(d)>0,\quad m\le c_2(d)\frac{\prod n_j}{\max\{n_j\}}. $$ Sharp bounds follow for classes of dominated mixed smoothness: $$ Θ_m(W^{(r,r,\ldots,r)}_p,L_q[0,1]^d)\asymp m^{-\frac{rd}{d-1}},\quad\mbox 2\le p\le\infty,\; 1\le q\le 2. $$

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↗

arXiv2021

Search of fractal space-filling curves with minimal dilation

We introduce an algorithm for a search of extremal fractal curves in large curve classes. It heavily uses SAT-solvers~ -- heuristic algorithms that find models for CNF boolean formulas. Our algorithm was implemented and applied to the search of fractal surjective curves $γ\colon[0,1]\to[0,1]^d$ with minimal dilation $$ \sup_{t_1<t_2}\frac{\|γ(t_2)-γ(t_1)\|^d}{t_2-t_1}. $$ We report new results of that search in the case of Euclidean norm. We have found a new curve that we call "YE", a self-similar (monofractal) plane curve of genus $5\times 5$ with dilation $5\frac{43}{73}=5.5890\ldots$. In dimension $3$ we have found facet-gated bifractals (that we call "Spring") of genus $2\times2\times 2$ with dilation $<17$. In dimension $4$ we obtained that there is a curve with dilation $<62$. Some lower bounds on the dilation for wider classes of cubically decomposable curves are proved.

math.MG↗

arXiv2020

Kolmogorov widths of Besov classes $B^1_{1,θ}$ and products of octahedra

In this paper we find the orders of decay for Kolmogorov widths of some Besov classes related to $W^1_1$ (the behaviour of the widths for $W^1_1$ remains unknown): $$ d_n(B^1_{1,θ}[0,1],L_q[0,1])\asymp n^{-1/2}\log^{\max(\frac12,1-\frac{1}θ)}n,\quad 2<q<\infty. $$ The proof relies on the lower bound for widths of product of octahedra in a special norm (maximum of two weighted $\ell_q$ norms). This bound generalizes the theorem of B.S.~Kashin on widths of octahedra in $\ell_q^N$.

math.FA↗