Estimating Renyi Entropy of Discrete Distributions

It was shown recently that estimating the Shannon entropy H(p) of a discrete k-symbol distribution p requires Θ(k/log k) samples, a number that grows near-linearly in the support size. In many applications, H(p) can be replaced by the more general Rényi entropy of order α and H α (p). We determine...

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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 63; no. 1; pp. 38 - 56
Main Authors: Acharya, Jayadev, Orlitsky, Alon, Suresh, Ananda Theertha, Tyagi, Himanshu
Format: Journal Article
Language:English
Published: New York IEEE 01.01.2017
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Additives
Approximation
Complexity theory
Electronic mail
Entropy
Entropy (Information theory)
Entropy estimation
Estimating techniques
Estimation
Functions (mathematics)
Genetics
Lower bounds
Mathematical analysis
minimax lower bounds
Polynomials
sample complexity
sublinear algorithms
Upper bound
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Summary:It was shown recently that estimating the Shannon entropy H(p) of a discrete k-symbol distribution p requires Θ(k/log k) samples, a number that grows near-linearly in the support size. In many applications, H(p) can be replaced by the more general Rényi entropy of order α and H α (p). We determine the number of samples needed to estimate H α (p) for all α, showing that α <; 1 requires a super-linear, roughly k 1/α samples, noninteger α > 1 requires a near-linear k samples, but, perhaps surprisingly, integer α > 1 requires only Θ(k 1-1/α ) samples. Furthermore, developing on a recently established connection between polynomial approximation and estimation of additive functions of the form Σ x f (p x ), we reduce the sample complexity for noninteger values of α by a factor of log k compared with the empirical estimator. The estimators achieving these bounds are simple and run in time linear in the number of samples. Our lower bounds provide explicit constructions of distributions with different Rényi entropies that are hard to distinguish.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2016.2620435