Entropy estimate for high-dimensional monotonic functions
By Fuchang Gao, Jon A. Wellner. 2005
61 citations recorded by the source corpus.
We establish upper and lower bounds for the metric entropy and bracketing entropy of the class of d-dimensional bounded monotonic functions under L^p norms. It is interesting to see that both the metric entropy and bracketing entropy have different behaviors for pd/(d-1). We apply the new bounds for bracketing entropy to establish a global rate of convergence of the MLE of a d-dimensional monotone density.
Subfields: Applied Mathematics, Computational Mechanics, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/a8b36e22c04e87f8bfffd417f5024cb950c113d0
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In Applied Mathematics
In Computational Mechanics
- Adityanand Guntuboyina
- Bodhisattva Sen
- Sabyasachi Chatterjee
- Pierre C. Bellec
- Richard J. Samworth
- Philippe Rigollet
- Nicolas Flammarion
- Cheng Mao
Citations in the corpus
- On matrix estimation under monotonicity constraints (2015, background): “However existing metric entropy results for classes of bivariate nondecreasing functions (as in Gao and Wellner [12]) require the functions to be uniformly bounded.”
- Isotonic regression in general dimensions (2017, unknown): “ n −f0k 2 L2(Q) ≤ sup d Ekfˆ n − f0k 2 L2(Q) ≤ Cdn −1/d log4 n, where the infimum is taken over all measurable functions of {Y(x) : x∈ n−1 1 Ld,n}. This corollary is surprising in the following sense. Gao and Wellner (2007, Theorem 1.1) proved that for d≥ 3, logN ε,Fd ∩ B∞(1),k·k2 ≍d ε −2(d 1). (9) I”