arXiv:1902.06120v2 [cs.IT] 2 Jun 2019 Transportation Proofs of Rényi Entropy Power Inequalities Olivier Rioul LTCI, Télécom ParisTech, 75013, Paris, France Institut Polytechnique de Paris Email: olivier.rioul@telecom-paristech.fr Abstract—A framework for deriving Rényi entropy-power inequalities (EPIs) is presented that uses linearization and an inequality of Dembo, Cover, and Thomas. Simple arguments are given to recover the previously known Rényi EPIs and derive new ones, by unifying a multiplicative form with con- stant c and a modification with exponent α of previous works. An information-theoretic proof of the Dembo-Cover-Thomas inequality—equivalent to Young’s convolutional inequality with optimal constants—is provided, based on properties of Rényi conditional and relative entropies and using transportation ar- guments from Gaussian densities. For log-concave densities, a transportation proof of a sharp varentropy bound is presented. I. I NTRODUCTION Throughout this paper we consider n-dimensional zero- mean random vectors X ∈ R n having densities. If X ∼ f has density f ∈ L r (R n ) where r> 0 and r =1, its Rényi entropy of exponent r (or r-entropy) is h r (X )= 1 1 − r log  R n f r (x)dx = −r ′ log ‖f ‖ r (1) where ‖f ‖ r denotes the L r norm of f , and r ′ = r r−1 is the conjugate exponent of r, such that 1 r + 1 r ′ =1. Notice two distinct situations: either r> 1 and r ′ > 1, or 0 <r< 1 and r ′ < 0. It is known that the limit as r → 1 is the Shannon (dif- ferential) entropy h 1 (X )= h(X )= −  R n f (x) log f (x)dx. Letting N (X ) = exp ( 2h(X )/n ) be the corresponding entropy power, the famous entropy power inequality (EPI) can be written in the form N  m  i=1 X i  ≥ m  i=1 N (X i ) (2) for any independent random vectors X 1 ,X 2 ,...,X m ∈ R n . The EPI dates back to Shannon’s seminal paper [1] and has a long history [2]. The link with the Rényi entropy h r (X ) was first made by Dembo, Cover and Thomas [3] in connection with Young’s convolutional inequality with sharp constants, where Shannon’s EPI is obtained by letting exponents r → 1 [4, Thm 17.8.3]. Recently, there has been increasing interest in Rényi entropy-power inequalities [5]. The Rényi entropy-power it- self was first defined in [6]. We follow a slightly different definition [7] where, as in Shannon’s original definition [1], the Rényi entropy-power N r (X ) equals (up to a multiplicative constant) the average power of a white Gaussian vector having the same Rényi entropy as X —hence the name “entropy power”. If X ∗ ∼N (0,σ 2 I) is white Gaussian, an easy calculation yields h r (X ∗ )= n 2 log(2πσ 2 )+ n 2 r ′ log r r . (3) Since equating h r (X ∗ )= h r (X ) gives σ 2 = e 2hr (X)/n 2πr r ′ /r , we define the Rényi entropy power as N r (X )= e 2hr(X)/n . (4) Bobkov and Chistyakov [7] extended the classical Shannon’s EPI (2) to the Rényi entropy by incorporating a multiplicative constant c> 0 that depends on r: N r  m  i=1 X i  ≥ c m  i=1 N r (X i ). (5) Ram and Sason [8] improved (increased) the value of c by making it depend also on the number m of independent vectors X 1 ,X 2 ,...,X m . Bobkov and Marsiglietti [9] proved another modification of the EPI for the Rényi entropy: N r α  m  i=1 X i  ≥ m  i=1 N r α (X i ) (6) with a power exponent parameter α> 0 whose value was further improved (decreased 1 ) by Li [10]. All the above EPIs were found for Rényi entropies of orders r>1. Recently, the α-modification of the Rényi EPI (6) was extended to orders <1 for two independent variables having log-concave densities by Marsiglietti and Melbourne [11]. The starting point of all the above works was Young’s strength- ened convolutional inequality. In this paper, we build on the results of [12] that provides a comprehensive framework with simple proofs for Rényi EPIs of the general form N r α  m  i=1 X i  ≥ c m  i=1 N r α (X i ) (7) with constant c> 0 and exponent α> 0. The framework uses only basic properties of Rényi entropies and is based on a transportation argument from normal densities and a change of variable by rotation, which had been previously used to give a simple proof of Shannon’s original EPI [13]. 1 Due to the non-increasing property of the α-norm, if (6) holds for α it also holds for any α ′ >α.