Estimates of entropy numbers in probabilistic setting

Abstract In this paper, we define the entropy number in probabilistic setting and determine the exact order of entropy number of finite-dimensional space in probabilistic setting. Moreover, we also estimate the sharp order of entropy number of univariate Sobolev space in probabilistic setting by discretization method.


Introduction
Entropy numbers are closely related to Kolmogorov's concept of metric entropy, which dates back to the 1930s. Basic properties of entropy numbers may be found in the monographs by Pietsch [1], Carl and Stephani [2], and Lorentz et al. [3]. Schütt [4], Edmunds and Triebel [5], and Kühn [6] have determined the asymptotic behavior of the entropy numbers of identity operators and diagonal operators under mild regularity and decay conditions on the generating sequence. Dung [7] investigated optimal entropy numbers of multivariate periodic functions with mixed smoothness. Belinsky [8] obtained estimates for the entropy numbers of classes of functions with conditions on the mixed derivative in the uniform and integral metrics.
In [9], an approximation criterion called the probabilistic criterion was considered that permits one to construct the distribution function with respect to a given measure for the best approximation functional. In [10], Maiorov introduced the concept of Kolmogorov ( ) n δ , -widths, i.e., Kolmogorov-widths in probabilistic setting, and studied the Kolmogorov ( ) n δ , -widths of univariate Sobolev space ( ) W r 2 with the Gaussian measure μ. In particular, he also gave some beautiful results of Kolmogorov ( ) n δ , -widths of the finite-dimensional space R m equipped with the standard Gaussian measure. In [11], Chen determined the asymptotic order of the Kolmogorov ( ) n δ , -widths of the multivariate Sobolev space with mixed derivative ( ) . Motivated by the aforementioned studies, in this paper, we introduce the concept of ( ) n δ , -entropy numbers which is entropy numbers in probabilistic setting. In the probabilistic setting, ( ) n δ , -entropy number is defined as in the worst case setting, but disregarding a set of measures at most δ, where ∈ [ ] δ 0, 1 . If disregarding a null set, then ( ) n δ , -entropy number equals entropy number. This concept is an analogue to the Kolmogorov ( ) n δ , -widths and it generalizes the concept of entropy numbers. Then we give the asymptotic order of the entropy numbers in probabilistic setting of the finite-dimensional space m equipped with the standard Gaussian measure in l q m -metric, ≤ ≤ q 1 2. In fact, this asymptotic quantity has the same order with Kolmogorov ( ) n δ , -widths of the finite-dimensional space. Moreover, using the discretization method we study the entropy numbers in probabilistic setting of Sobolev space 2.

Preliminaries
We first give some notions. We use to denote the set of integer numbers. Let 0 . Assume that c, c i , = … i 0, 1, , are positive constants depending only on the parameters p, q, r, ρ. For two positive functions ( ) a y and ( ) b y , ∈ y D, we write ( ) ≍ ( ) a y b y or ( ) ≪ ( ) a y b y if there exists constants c, c 1 , and c 2 such that ≤ ( )/ ( ) ≤ c a y b y c 1 2 or ( ) ≤ ( ) a y cb y for any ∈ y D. Next, we recall some definitions. Let W and M be subsets of a normed linear space X, the quantity sup inf , and | | M denotes the cardinality of M. Detailed information about the usual entropy numbers may be found in [2,5]. Assume that W contains a Borel field B consisting of open subsets of W and is equipped with a probability measure μ defined on B, i.e., μ is a σ-additive non-negative function on B, and μ(W) = 1.
Let ∈ ( ] δ 0, 1 be an arbitrary number. The ( ) n δ , -entropy number of a set W with a measure μ in the space X is defined by , satisfy the inequalities where ′ c p and ″ c p depend only on p. This follows from the relation where Γ is the Euler Γ-function. .
be a non-increasing sequence of non-negative numbers and let , , m m We consider in m the standard Gaussian measure = ν ν m , which is defined on Borel subsets ⊆ G R m by , and ∈ ( ] δ 0, 1 be arbitrary, we define ( ) n δ , -entropy numbers of the space m equipped with the Gaussian measure v in the space l q m : where the infimum is taken over all possible Borel subsets ⊂ The rest of the paper is organized as follows. In Section 3, we determine the asymptotic order of the ( ) n δ , -entropy numbers of finite-dimensional set. In Section 4, we calculate the asymptotic order of the Now we state our main result.
, -widths of finite-dimensional set was obtained by Maiorov [10]. If ≤ n m, we can see that these two numbers are asymptotically equivalent when ≤ ≤ q 1 2.
To prove Theorem 1, we need some auxiliary assertions.
where c is an absolute constant.
Lemma 1 follows immediately from Proposition 1, here we omit the proof. We first find some simple Borel subsets ⊂ . The following lemma provides such subsets G.

Lemma 2. [12]
There exists an absolute positive constant c 0 such that for any ∈ ( / ] δ 0, 1 2 , we have Proof of Theorem 1. We first establish the upper estimate of ( ) ε ν l , , Estimates of entropy numbers in probabilistic setting  1637 Now we estimate the lower bound of For any ⊆ G m , it holds that ( ) = ∈ ( / ] ν G δ 0, 1 2 . We consider the following two cases: is unbounded. In this case, it is trivial that . From this t 1 must exist. Maiorov gives the following result in [10]: m a x , l n1 . It is easy to verify that ( ) = ( ) which together with (5) gives Therefore, By the definition of G t1 , D 1 , and D 2 , we have According to (1.2.1) (see [2, p. 10]), we get It follows from (6) and (7) that where c depends only on q. Using (3) and (9), we have , continuously. In this paper, we suppose > / r 1 2. Let the space ( ) More detailed information about the Gaussian measure in Banach space is contained in the books of [13,14]. Now we state our main result of this section.
, -entropy numbers of ( ) W r 2 with the Gaussian measure μ in the space ( ) L q satisfy asymptotic relation , -widths of infinite-dimensional set was obtained by Maiorov [10]. We can see that these two numbers are asymptotically equivalent when < ≤ q 1 2.
To prove Theorem 2, we need two auxiliary discretization theorems (Theorems 3 and 4) that reduce the computation of ( ) N δ , -entropy numbers . We introduce some notations and lemmas.
The following two known lemmas are crucial for establishing discretization theorems (Theorems 3 and 4).
Moreover, the following relation is true: where the constant in the equivalence does not depend on k. From the aforementioned inequality and (10), we have L kr r L kr k q r j j l kr k q r j j l q q q m k q m k (11) For any ∈ + k , we consider a mapping It follows from Lemma 4 that I k is linear isomorphic from the space F m k to the space m k . Hence, there exists a constant c 1 such that . Then by virtue of (11) there exists constant c 2 independent of k such that , which is the sum of the subset . From the hypothesis of the theorem, we get Consequently, by the definitions of ( ( ) ( )) ε W μ L , , where ≔ ( ( ) ( )) ε ε W μ L , ,