This survey addresses sampling discretization and its connections with other areas of mathematics. We present here known results on sampling discretization of both integral norms and the uniform norm beginning with classical results and ending with v
ery recent achievements. We also show how sampling discretization connects to spectral properties and operator norms of submatrices, embedding of finite-dimensional subspaces, moments of marginals of high-dimensional distributions, and learning theory. Along with the corresponding results, important techniques for proving those results are discussed as well.
In this paper we study bounds for the total variation distance between two second degree polynomials in normal random variables provided that they essentially depend on at least three variables.
We study the regularity of densities of distributions that are polynomial images of the standard Gaussian measure on $mathbb{R}^n$. We assume that the degree of a polynomial is fixed and that each variable enters to a power bounded by another fixed number.
The paper studies the sampling discretization problem for integral norms on subspaces of $L^p(mu)$. Several close to optimal results are obtained on subspaces for which certain Nikolskii-type inequality is valid. The problem of norms discretization i
s connected with the probabilistic question about the approximation with high probability of marginals of a high dimensional random vector by sampling. As a byproduct of our approach we refine the result of O. Gu$acute{e}$don and M. Rudelson concerning the approximation of marginals. In particular, the obtained improvement recovers a theorem of J. Bourgain, J. Lindenstrauss, and V. Milman concerning embeddings of finite dimensional subspaces of $L^p[0, 1]$ into $ell_p^m$. The proofs in the paper use the recent developments of the chaining technique by R. van Handel.
In this paper we study the regularity properties of linear and polynomial images of Skorohod differentiable measures. Firstly, we obtain estimates for the Skorohod derivative norm of a projection of a product of Scorohod differentiable measures. In t
he second part of the paper we prove Nikolskii--Besov regularity of a polynomial image of a Skorohod differentiable measure on $mathbb{R}^n$.
We study fractional smoothness of measures on $mathbb{R}^k$, that are images of a Gaussian measure under mappings from Gaussian Sobolev classes. As a consequence we obtain Nikolskii--Besov fractional regularity of these distributions under some weak nondegeneracy assumption.
The paper provides an estimate of the total variation distance between distributions of polynomials defined on a space equipped with a logarithmically concave measure in terms of the $L^2$-distance between these polynomials.
Let $gamma$ be the standard Gaussian measure on $mathbb{R}^n$ and let $mathcal{P}_{gamma}$ be the space of probability measures that are absolutely continuous with respect to $gamma$. We study lower bounds for the functional $mathcal{F}_{gamma}(mu) =
{rm Ent}(mu) - frac{1}{2} W^2_2(mu, u)$, where $mu in mathcal{P}_{gamma}, u in mathcal{P}_{gamma}$, ${rm Ent}(mu) = int logbigl( frac{mu}{gamma}bigr) d mu$ is the relative Gaussian entropy, and $W_2$ is the quadratic Kantorovich distance. The minimizers of $mathcal{F}_{gamma}$ are solutions to a dimension-free Gaussian analog of the (real) Kahler-Einstein equation. We show that $mathcal{F}_{gamma}(mu) $ is bounded from below under the assumption that the Gaussian Fisher information of $ u$ is finite and prove a priori estimates for the minimizers. Our approach relies on certain stability estimates for the Gaussian log-Sobolev and Talagrand transportation inequalities.
We give a new description of classical Besov spaces in terms of a new modulus of continuity. Then a similar approach is used to introduce Besov classes on an infinite-dimensional space endowed with a Gaussian measure.
We give a new characterization of Nikolskii-Besov classes of functions of fractional smoothness by means of a nonlinear integration by parts formula in the form of a nonlinear inequality. A similar characterization is obtained for Nikolskii-Besov cla
sses with respect to Gaussian measures on finite- and infinite-dimensional spaces.
