Entropy Estimate for $k$-Monotone Functions via Small Ball Probability of Integrated Brownian Motions

Gao, Fuchang

doi:10.1214/ecp.v13-1357

Cited by 17 publications

(13 citation statements)

References 18 publications

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“…In the case d = 1, Gao [11] removed the requirement of uniform Lipschitz condition, and obtained sharp bounds for log N (ε, C ∞ ([a, b]), L 2 ), and, in fact, provided upper and lower bounds of the order ε −1/k for the "k-monotone" classes…”

Section: Introductionmentioning

confidence: 99%

Entropy of Convex Functions on $$\mathbb {R}^d$$ R d

Gao

Wellner

2017

Constr Approx

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Let Ω be a bounded closed convex set in R d with non-empty interior, and let Cr(Ω) be the class of convex functions on Ω with L r -norm bounded by 1. We obtain sharp estimates of the ε-entropy of Cr(Ω) under L p (Ω) metrics, 1 ≤ p < r ≤ ∞. In particular, the results imply that the universal lower bound ε −d/2 is also an upper bound for all d-polytopes, and the universal upper bound of ε − (d−1) 2 · pr r−p for p > dr d+(d−1)r is attained by the closed unit ball. While a general convex body can be approximated by inscribed polytopes, the entropy rate does not carry over to the limiting body. Our results have applications to questions concerning rates of convergence of nonparametric estimators of high-dimensional shape-constrained functions.

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Section: Introductionmentioning

confidence: 99%

Entropy of Convex Functions on $$\mathbb {R}^d$$ R d

Gao

Wellner

2017

Constr Approx

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“…Using the L 2 estimate from Theorem 3 for the left-hand side, this shows This holds for all ε > 0. Letting ε tend to zero yields the lower bound for the entropy numbers, since the constant d α,β (defined in (9)) is continuous in the parameters.…”

Section: And Consequentlymentioning

confidence: 99%

Small Deviations for a Family of Smooth Gaussian Processes

et al. 2011

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We study the small deviation probabilities of a family of very smooth self-similar Gaussian processes. The canonical process from the family has the same scaling property as standard Brownian motion and plays an important role in the study of zeros of random polynomials.Our estimates are based on the entropy method, discovered in Kuelbs and Li ( The results are further applied to the entropy of function classes, generalizing results of Gao et al. (2010).

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“…Firstly, because k -monotone C ∞ functions are dense in

{scriptM}_{k}^{1} ([0, 1])

(cf. [8]), we can and will assume that all the densities in

{scriptM}_{k}^{1} ([0, 1])

are continuously k -times differentiable. Secondly, if for I = [ a, b ] ⊂ [0, 1] we define

{scriptH}_{k}^{B} (I) = {f : f (u) = g (b - u), g \in {scriptM}_{k}^{B} (I)} .

then for every

f \in {scriptH}_{k}^{1} ([0, 1])

\frac{d^{j} f (u)}{d u^{j}} = \frac{d^{j}}{d u^{j}} (g (1 - u)) = {(- 1)}^{j} g^{(j)} (1 - u) \geq 0

for all 0 ≤ j ≤ k , and for all u ∈ [0, 1].…”

Section: Under Hellinger Distancementioning

confidence: 99%

“…For k > 1, Gao[8] also established the following metric entropy bound for

{scriptM}_{k}^{B} ([0, 1])

C_{1} B^{1 ∕ k} ε^{- 1 ∕ k} \leq \log N (ε, {scriptM}_{k}^{B} ([0, 1]), {‖ \cdot ‖}_{2}) \leq C_{2} B^{1 ∕ k} ε^{- 1 ∕ k} .

The method revealed a nice connection between the metric entropy of these function classes and the small ball probability of k -times integrated Brownian motions. However, because for k > 1 the square root of a k -monotone function may not be k -monotone, the metric entropy estimate under L 2 distance does not yield an estimate under the Hellinger distance.…”