DOCSLIB.ORG

Rényi Entropy Power and Normal Transport

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Rényi Entropy Power and Normal Transport ISITA2020, Kapolei, Hawai’i, USA, October 24-27, 2020 Rényi Entropy Power and Normal Transport Olivier Rioul LTCI, Télécom Paris, Institut Polytechnique de Paris, 91120, Palaiseau, France Abstract—A framework for deriving Rényi entropy-power with a power exponent parameter ↵>0. Due to the non- inequalities (REPIs) is presented that uses linearization and an increasing property of the ↵-norm, if (4) holds for ↵ it also inequality of Dembo, Cover, and Thomas. Simple arguments holds for any ↵0 >↵. The value of ↵ was further improved are given to recover the previously known Rényi EPIs and derive new ones, by unifying a multiplicative form with con- (decreased) by Li [9]. All the above EPIs were found for Rényi stant c and a modification with exponent ↵ of previous works. entropies of orders r>1. Recently, the ↵-modification of the An information-theoretic proof of the Dembo-Cover-Thomas Rényi EPI (4) was extended to orders <1 for two independent inequality—equivalent to Young’s convolutional inequality with variables having log-concave densities by Marsiglietti and optimal constants—is provided, based on properties of Rényi Melbourne [10]. The starting point of all the above works was conditional and relative entropies and using transportation ar- guments from Gaussian densities. For log-concave densities, a Young’s strengthened convolutional inequality. transportation proof of a sharp varentropy bound is presented. In this paper, we build on the results of [11] to provide This work was partially presented at the 2019 Information Theory simple proofs for Rényi EPIs of the general form and Applications Workshop, San Diego, CA. ↵ m m ↵ Nr Xi c Nr (Xi) (5) i=1 ≥ i=1 I. INTRODUCTION with constant c>⇣X 0 and⌘ exponentX ↵>0. The present We consider the r-entropy (Rényi entropy of exponent framework uses only basic properties of Rényi entropies and r, where r>0 and r =1) of a n-dimensional zero-mean is based on a transportation argument from normal densities 6 random vector X Rn having density f Lr(Rn): and a change of variable by rotation, which was previously 2 2 1 r used to give a simple proof of Shannon’s original EPI [12]. hr(X)= log f (x)dx = r0 log f r (1) 1 r n R − k k II. LINEARIZATION − Zr r where f denotes the L norm of f, and r0 = is the k kr r 1 The first step toward proving (5) is the following linearization conjugate exponent of r, such that 1 + 1 =1. Notice that− either r r0 lemma which generalizes [9, Lemma 2.1]. r>1 and r0 > 1, or 0 <r<1 and r0 < 0. The limit as r 1 ! For independent X1,X2,...,Xm, the Rényi EPI is the classical h1(X)=h(X)= n f(x) log f(x)dx. Lemma 1. − R Letting N(X)=exp 2h(X)/n be the corresponding entropy in the general form (5) is equivalent to the following inequality R m m power [1], the famous entropy power inequality (EPI) [1], [2] n log c 1 m m hr pλiXi λihr(Xi) 2 ↵ + ↵ 1 H(λ) writes N i=1 Xi i=1 N(Xi) for any independent i=1 − i=1 ≥ − ≥ n for any distribution λ =(λ ,...,λ ) of entropy H(λ). (6) random vectors X1,X2,...,Xm R . The link with the P P 1 m ⇣P ⌘ P 2 Rényi entropy hr(X) was first made in [3] in connection with Proof. Note the scaling property h (aX)=h (X)+n log a r r | | a strengthened Young’s convolutional inequality, where the for any a =0, established by a change of variable. It follows 6 2 EPI is obtained by letting exponents tend to 1 [4, Thm 17.8.3]. that Nr(aX)=a Nr(X). Now first suppose (5) holds. Then Recently, there has been increasing interest in Rényi entropy- m n ↵ m hr pλiXi = log Nr pλiXi (7) power inequalities [5]. The Rényi entropy-power Nr(X) is i=1 2↵ i=1 n log m N ↵(pλ X )+ n log c defined [6] as the average power of a white Gaussian vector P 2↵ i=1 r i i P2↵ 2 ≥ m ↵ having the same Rényi entropy as X. If X⇤ (0,σ I) is = n log λ↵N (X )+ n log c (8) ⇠N 2↵ Pi=1 i r i 2↵ white Gaussian, an easy calculation yields n m ↵ 1 ↵ n 2↵ i=1Pλi log λi − Nr (Xi) + 2↵ log c (9) n 2 n log r ≥ hr(X⇤)= log(2⇡ )+ r0 . (2) m n(↵ 1) m n 2 2 r = iP=1 λihr(Xi)+ 2↵− i=1 λi log λi + 2↵ log c 2 e2hr (X)/n Since equating hr(X⇤)=hr(X) gives σ = , we which proves (6). The scaling property is used in (8) and the 2⇡rr0/r P P 2hr (X)/n define Nr(X)=e as the r-entropy power. concavity of the logarithm is used in (9). Bobkov and Chistyakov [6] extended the classical EPI to Conversely, suppose that (6) is satisfied for all λi > 0 such m ↵ m ↵ the r-entropy by incorporating a r-dependent constant c>0: that i=1 λi =1. Set λi = Nr (Xi)/ i=1 Nr (Xi). Then ↵ m m m 2↵ m Xi N Xi =exp hr pλi r P i=1 n i=1 Ppλi Nr Xi c Nr(Xi). (3) m 1 i=1 ≥ i=1 2↵ m X (1 ↵) λi log i − i=1 λi ⇣ ⌘ Pexp n i=1 λihr P c e Ram and SasonX [7] improved (increased)X the value of c by ≥ pλi ⇥ · m ⇣ ⌘λi m P λi making it depend also on the number m of independent vectors P↵ Xi ↵ 1 ↵ 1 = c N λ − = c N (X )λ− r pλ i r i i X1,X2,...,Xm. Bobkov and Marsiglietti [8] proved another i=1 i i=1 ⇣ ⇣ ⌘ m⌘ ⇣ ⌘ modification of the EPI for the Rényi entropy: Q m ↵ i=1 λi Q m ↵ = c i=1 Nr (Xi) = c i=1 Nr (Xi) ↵ m m ↵ P Nr Xi Nr (Xi) (4) which proves (5). i=1 ≥ i=1 P P ⇣X ⌘ X Copyright (C) 2020 by IEICE 1 ISITA2020, Kapolei, Hawai’i, USA, October 24-27, 2020 III. THE REPI OF DEMBO-COVER-THOMAS Proof. By Lemma 1 for ↵ =1we only need to check that n As a second ingredient we have the following result, which the r.h.s. of (13) is greater than 2 log c for any choice of was essentially established by Dembo, Cover and Thomas [3]. the λi’s, that is, for any choice of exponents ri such that m 1 1 = . Thus, (3) will hold for log c =minλ A(λ). It is this Rényi version of the EPI which led them to prove i=1 ri0 r0 Shannon’s original EPI by letting Rényi exponents 1. Now, by the log-sum inequality [4, Thm 2.7.1], P m ! m 1 1 m 1 1 m 1 i=1 ri 1 r0 Theorem 1. Let r1,...,rm,r be exponents those conjugates log log =(m r ) log − m 1 1 ri ri ≥ ri m − 0 m r10 ,...,rm0 ,r0 are of the same sign and satisfy i=1 r = r i=1 i=1 P i0 0 X X (15) and let λ1,...,λm be the discrete probability distribution with equality if and only if all r are equal, that is, the r0 P i λi = . Then, for independent zero-mean X1,X2,...,Xm, log r ri0 λ are equal to 1/m. Thus, min A(λ)=r0 +(m m m i λ r − h pλ X λ h (X ) m 1/r0 r i i i ri i 1/r0) log −m = log c. i=1 − i=1 ⇥ ⇣ ⌘ m m (10) log r 1 P P Note that log⇤c = r0 +(mr0 1) log 1 < 0 p r mr0 hr λiXi⇤ λihri (Xi⇤) log r − − ≥ i=1 − i=1 decreases (and tends to r0 1) as m increases. Thus, a ⇣ ⌘ r − where X1⇤,X2⇤,...,Xm⇤ Pare i.i.d. standardP Gaussian (0, I). universal constant independent of m is obtained by taking N r0/r Equality holds if and only if the Xi are i.i.d. Gaussian. 1 mr0 1 r c =inf rr0/r 1 − = , (16) m − mr0 e It is easily seen from the expression (2) of the Rényi as was established by Bobkov and Chistyakov [6]. entropy of a Gaussian that (10) is equivalent to m m m n log r log ri Proposition 2 (Li [9]). The Rényi EPI (4) holds for r>1 hr pλiXi λihr (Xi) r0 . (11) log r 1 i 2 r ri 2 1 − − ≥ − and ↵ = 1+r0 +(2r0 1) log2 1 . i=1 i=1 i=1 r − − 2r0 ⇣P ⌘ P ⇣ P ⌘ Note that the l.h.s. is very similar to that of (6) except that Li [9] remarked⇥ that this value of ↵ is strictly smaller⇤ (better) r+1 different Rényi exponents are present. This will be the crucial than the value ↵ = 2 obtained previously by Bobkov and step toward proving (5). Marsiglietti [8]. In [11] it is shown that it cannot be further Theorem 1 (for m =2) was derived in [3] as a rewriting improved in our framework by making it depend on m. of Young’s strengthened convolutional inequality with optimal Proof. Since the announced ↵ does not depend on m, we constants. Section VII provides a simple transportation proof, can always assume that m =2. By Lemma 1 for c =1, which uses only basic properties of Rényi entropies.
Load more

Recommended publications
  • On Relations Between the Relative Entropy and $\Chi^ 2$-Divergence
    On Relations Between the Relative Entropy and χ2-Divergence, Generalizations and Applications Tomohiro Nishiyama Igal Sason Abstract The relative entropy and chi-squared divergence are fundamental divergence measures in information theory and statistics. This paper is focused on a study of integral relations between the two divergences, the implications of these relations, their information-theoretic applications, and some generalizations pertaining to the rich class of f-divergences. Applications that are studied in this paper refer to lossless compression, the method of types and large deviations, strong data-processing inequalities, bounds on contraction coefficients and maximal correlation, and the convergence rate to stationarity of a type of discrete-time Markov chains. Keywords: Relative entropy, chi-squared divergence, f-divergences, method of types, large deviations, strong data-processing inequalities, information contraction, maximal correlation, Markov chains. I. INTRODUCTION The relative entropy (also known as the Kullback–Leibler divergence [28]) and the chi-squared divergence [46] are divergence measures which play a key role in information theory, statistics, learning, signal processing and other theoretical and applied branches of mathematics. These divergence measures are fundamental in problems pertaining to source and channel coding, combinatorics and large deviations theory, goodness-of-fit and independence tests in statistics, expectation-maximization iterative algorithms for estimating a distribution from an incomplete
    [Show full text]
  • A Lower Bound on the Differential Entropy of Log-Concave Random Vectors with Applications
    entropy Article A Lower Bound on the Differential Entropy of Log-Concave Random Vectors with Applications Arnaud Marsiglietti 1,* and Victoria Kostina 2 1 Center for the Mathematics of Information, California Institute of Technology, Pasadena, CA 91125, USA 2 Department of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125, USA; [email protected] * Correspondence: [email protected] Received: 18 January 2018; Accepted: 6 March 2018; Published: 9 March 2018 Abstract: We derive a lower bound on the differential entropy of a log-concave random variable X in terms of the p-th absolute moment of X. The new bound leads to a reverse entropy power inequality with an explicit constant, and to new bounds on the rate-distortion function and the channel capacity. Specifically, we study the rate-distortion function for log-concave sources and distortion measure ( ) = j − jr ≥ d x, xˆ x xˆ , with r 1, and we establish thatp the difference between the rate-distortion function and the Shannon lower bound is at most log( pe) ≈ 1.5 bits, independently of r and the target q pe distortion d. For mean-square error distortion, the difference is at most log( 2 ) ≈ 1 bit, regardless of d. We also provide bounds on the capacity of memoryless additive noise channels when the noise is log-concave. We show that the difference between the capacity of such channels and the capacity of q pe the Gaussian channel with the same noise power is at most log( 2 ) ≈ 1 bit. Our results generalize to the case of a random vector X with possibly dependent coordinates.
    [Show full text]
  • Entropy Power Inequalities the American Institute of Mathematics
    Entropy power inequalities The American Institute of Mathematics The following compilation of participant contributions is only intended as a lead-in to the AIM workshop “Entropy power inequalities.” This material is not for public distribution. Corrections and new material are welcomed and can be sent to [email protected] Version: Fri Apr 28 17:41:04 2017 1 2 Table of Contents A.ParticipantContributions . 3 1. Anantharam, Venkatachalam 2. Bobkov, Sergey 3. Bustin, Ronit 4. Han, Guangyue 5. Jog, Varun 6. Johnson, Oliver 7. Kagan, Abram 8. Livshyts, Galyna 9. Madiman, Mokshay 10. Nayar, Piotr 11. Tkocz, Tomasz 3 Chapter A: Participant Contributions A.1 Anantharam, Venkatachalam EPIs for discrete random variables. Analogs of the EPI for intrinsic volumes. Evolution of the differential entropy under the heat equation. A.2 Bobkov, Sergey Together with Arnaud Marsliglietti we have been interested in extensions of the EPI to the more general R´enyi entropies. For a random vector X in Rn with density f, R´enyi’s entropy of order α> 0 is defined as − 2 n(α−1) α Nα(X)= f(x) dx , Z 2 which in the limit as α 1 becomes the entropy power N(x) = exp n f(x)log f(x) dx . However, the extension→ of the usual EPI cannot be of the form − R Nα(X + Y ) Nα(X)+ Nα(Y ), ≥ since the latter turns out to be false in general (even when X is normal and Y is nearly normal). Therefore, some modifications have to be done. As a natural variant, we consider proper powers of Nα.
    [Show full text]
  • On the Entropy of Sums
    On the Entropy of Sums Mokshay Madiman Department of Statistics Yale University Email: [email protected] Abstract— It is shown that the entropy of a sum of independent II. PRELIMINARIES random vectors is a submodular set function, and upper bounds on the entropy of sums are obtained as a result in both discrete Let X1,X2,...,Xn be a collection of random variables and continuous settings. These inequalities complement the lower taking values in some linear space, so that addition is a bounds provided by the entropy power inequalities of Madiman well-defined operation. We assume that the joint distribu- and Barron (2007). As applications, new inequalities for the tion has a density f with respect to some reference mea- determinants of sums of positive-definite matrices are presented. sure. This allows the definition of the entropy of various random variables depending on X1,...,Xn. For instance, I. INTRODUCTION H(X1,X2,...,Xn) = −E[log f(X1,X2,...,Xn)]. There are the familiar two canonical cases: (a) the random variables Entropies of sums are not as well understood as joint are real-valued and possess a probability density function, or entropies. Indeed, for the joint entropy, there is an elaborate (b) they are discrete. In the former case, H represents the history of entropy inequalities starting with the chain rule of differential entropy, and in the latter case, H represents the Shannon, whose major developments include works of Han, discrete entropy. We simply call H the entropy in all cases, ˇ Shearer, Fujishige, Yeung, Matu´s, and others. The earlier part and where the distinction matters, the relevant assumption will of this work, involving so-called Shannon-type inequalities be made explicit.
    [Show full text]
  • Relative Entropy and Score Function: New Information–Estimation Relationships Through Arbitrary Additive Perturbation
    Relative Entropy and Score Function: New Information–Estimation Relationships through Arbitrary Additive Perturbation Dongning Guo Department of Electrical Engineering & Computer Science Northwestern University Evanston, IL 60208, USA Abstract—This paper establishes new information–estimation error (MMSE) of a Gaussian model [2]: relationships pertaining to models with additive noise of arbitrary d p 1 distribution. In particular, we study the change in the relative I (X; γ X + N) = mmse PX ; γ (2) entropy between two probability measures when both of them dγ 2 are perturbed by a small amount of the same additive noise. where X ∼ P and mmseP ; γ denotes the MMSE of It is shown that the rate of the change with respect to the X p X energy of the perturbation can be expressed in terms of the mean estimating X given γ X + N. The parameter γ ≥ 0 is squared difference of the score functions of the two distributions, understood as the signal-to-noise ratio (SNR) of the Gaussian and, rather surprisingly, is unrelated to the distribution of the model. By-products of formula (2) include the representation perturbation otherwise. The result holds true for the classical of the entropy, differential entropy and the non-Gaussianness relative entropy (or Kullback–Leibler distance), as well as two of its generalizations: Renyi’s´ relative entropy and the f-divergence. (measured in relative entropy) in terms of the MMSE [2]–[4]. The result generalizes a recent relationship between the relative Several generalizations and extensions of the previous results entropy and mean squared errors pertaining to Gaussian noise are found in [5]–[7].
    [Show full text]
  • Cramér-Rao and Moment-Entropy Inequalities for Renyi Entropy And
    1 Cramer-Rao´ and moment-entropy inequalities for Renyi entropy and generalized Fisher information Erwin Lutwak, Deane Yang, and Gaoyong Zhang Abstract— The moment-entropy inequality shows that a con- Analogues for convex and star bodies of the moment- tinuous random variable with given second moment and maximal entropy, Fisher information-entropy, and Cramer-Rao´ inequal- Shannon entropy must be Gaussian. Stam’s inequality shows that ities had been established earlier by the authors [3], [4], [5], a continuous random variable with given Fisher information and minimal Shannon entropy must also be Gaussian. The Cramer-´ [6], [7] Rao inequality is a direct consequence of these two inequalities. In this paper the inequalities above are extended to Renyi II. DEFINITIONS entropy, p-th moment, and generalized Fisher information. Gen- Throughout this paper, unless otherwise indicated, all inte- eralized Gaussian random densities are introduced and shown to be the extremal densities for the new inequalities. An extension of grals are with respect to Lebesgue measure over the real line the Cramer–Rao´ inequality is derived as a consequence of these R. All densities are probability densities on R. moment and Fisher information inequalities. Index Terms— entropy, Renyi entropy, moment, Fisher infor- A. Entropy mation, information theory, information measure The Shannon entropy of a density f is defined to be Z h[f] = − f log f, (1) I. INTRODUCTION R provided that the integral above exists. For λ > 0 the λ-Renyi HE moment-entropy inequality shows that a continuous entropy power of a density is defined to be random variable with given second moment and maximal T 1 Shannon entropy must be Gaussian (see, for example, Theorem Z 1−λ λ 9.6.5 in [1]).
    [Show full text]
  • Two Remarks on Generalized Entropy Power Inequalities
    TWO REMARKS ON GENERALIZED ENTROPY POWER INEQUALITIES MOKSHAY MADIMAN, PIOTR NAYAR, AND TOMASZ TKOCZ Abstract. This note contributes to the understanding of generalized entropy power inequalities. Our main goal is to construct a counter-example regarding monotonicity and entropy comparison of weighted sums of independent identically distributed log- concave random variables. We also present a complex analogue of a recent dependent entropy power inequality of Hao and Jog, and give a very simple proof. 2010 Mathematics Subject Classification. Primary 94A17; Secondary 60E15. Key words. entropy, log-concave, Schur-concave, unconditional. 1. Introduction The differential entropy of a random vector X with density f (with respect to d Lebesgue measure on R ) is defined as Z h (X) = − f log f; Rd provided that this integral exists. When the variance of a real-valued random variable X is kept fixed, it is a long known fact [11] that the differential entropy is maximized by taking X to be Gaussian. A related functional is the entropy power of X, defined 2h(X) by N(X) = e d : As is usual, we abuse notation and write h(X) and N(X), even though these are functionals depending only on the density of X and not on its random realization. The entropy power inequality is a fundamental inequality in both Information The- ory and Probability, stated first by Shannon [34] and proved by Stam [36]. It states d that for any two independent random vectors X and Y in R such that the entropies of X; Y and X + Y exist, N(X + Y ) ≥ N(X) + N(Y ): In fact, it holds without even assuming the existence of entropies as long as we set an entropy power to 0 whenever the corresponding entropy does not exist, as noted by [8].
    [Show full text]
  • Entropy Power Inequalities for Qudits M¯Arisozols University of Cambridge
    Entropy power inequalities for qudits M¯arisOzols University of Cambridge Nilanjana Datta Koenraad Audenaert University of Cambridge Royal Holloway & Ghent output X XYl Y l 2 [0, 1] – how much of the signal gets through How noisy is the output? ? H(X l Y) ≥ lH(X) + (1 − l)H(Y) Prototypic entropy power inequality... Additive noise channel noise Y input X X Yl Y l 2 [0, 1] – how much of the signal gets through How noisy is the output? ? H(X l Y) ≥ lH(X) + (1 − l)H(Y) Prototypic entropy power inequality... Additive noise channel noise Y input output X X X Xl Y l 2 [0, 1] – how much of the signal gets through How noisy is the output? ? H(X l Y) ≥ lH(X) + (1 − l)H(Y) Prototypic entropy power inequality... Additive noise channel noise Y input output X Y XY How noisy is the output? ? H(X l Y) ≥ lH(X) + (1 − l)H(Y) Prototypic entropy power inequality... Additive noise channel noise Y input output X l X l Y l 2 [0, 1] – how much of the signal gets through XY Additive noise channel noise Y input output X l X l Y l 2 [0, 1] – how much of the signal gets through How noisy is the output? ? H(X l Y) ≥ lH(X) + (1 − l)H(Y) Prototypic entropy power inequality... = convolution = beamsplitter = partial swap f (r l s) ≥ lf (r) + (1 − l)f (s) I r, s are distributions / states cH(·) I f (·) is an entropic function such as H(·) or e I r l s interpolates between r and s where l 2 [0, 1] Entropy power inequalities Classical Quantum Shannon Koenig & Smith Continuous [Sha48] [KS14, DMG14] This work Discrete — = convolution = beamsplitter = partial swap I
    [Show full text]
  • On Relations Between the Relative Entropy and Χ2-Divergence, Generalizations and Applications
    entropy Article On Relations Between the Relative Entropy and c2-Divergence, Generalizations and Applications Tomohiro Nishiyama 1 and Igal Sason 2,* 1 Independent Researcher, Tokyo 206–0003, Japan; [email protected] 2 Faculty of Electrical Engineering, Technion—Israel Institute of Technology, Technion City, Haifa 3200003, Israel * Correspondence: [email protected]; Tel.: +972-4-8294699 Received: 22 April 2020; Accepted: 17 May 2020; Published: 18 May 2020 Abstract: The relative entropy and the chi-squared divergence are fundamental divergence measures in information theory and statistics. This paper is focused on a study of integral relations between the two divergences, the implications of these relations, their information-theoretic applications, and some generalizations pertaining to the rich class of f -divergences. Applications that are studied in this paper refer to lossless compression, the method of types and large deviations, strong data–processing inequalities, bounds on contraction coefficients and maximal correlation, and the convergence rate to stationarity of a type of discrete-time Markov chains. Keywords: relative entropy; chi-squared divergence; f -divergences; method of types; large deviations; strong data–processing inequalities; information contraction; maximal correlation; Markov chains 1. Introduction The relative entropy (also known as the Kullback–Leibler divergence [1]) and the chi-squared divergence [2] are divergence measures which play a key role in information theory, statistics, learning, signal processing, and other theoretical and applied branches of mathematics. These divergence measures are fundamental in problems pertaining to source and channel coding, combinatorics and large deviations theory, goodness-of-fit and independence tests in statistics, expectation–maximization iterative algorithms for estimating a distribution from an incomplete data, and other sorts of problems (the reader is referred to the tutorial paper by Csiszár and Shields [3]).
    [Show full text]
  • On the Similarity of the Entropy Power Inequality And
    IEEE TRANSACTIONSONINFORMATIONTHEORY,VOL. IT-30,N0. 6,NOVEmER1984 837 Correspondence On the Similarity of the Entropy Power Inequality The preceeding equations allow the entropy power inequality and the Brunn-Minkowski Inequality to be rewritten in the equivalent form Mc4x H.M.COSTAmMBER,IEEE,AND H(X+ Y) 2 H(X’+ Y’) (4) THOMAS M. COVER FELLOW, IEEE where X’ and Y’ are independent normal variables with corre- sponding entropies H( X’) = H(X) and H(Y’) = H(Y). Verifi- Abstract-The entropy power inequality states that the effective vari- cation of this restatement follows from the use of (1) to show that ance (entropy power) of the sum of two independent random variables is greater than the sum of their effective variances. The Brunn-Minkowski -e2H(X+Y)1 > u...eeWX)1 + leWY) inequality states that the effective radius of the set sum of two sets is 27re - 27re 2ae greater than the sum of their effective radii. Both these inequalities are 1 recast in a form that enhances their similarity. In spite of this similarity, E---e ZH(X’) 1 1 e2H(Y’) there is as yet no common proof of the inequalities. Nevertheless, their / 27re 2?re intriguing similarity suggests that new results relating to entropies from = a$ + u$ known results in geometry and vice versa may be found. Two applications of this reasoning are presented. First, an isoperimetric inequality for = ux2 ’+y’ entropy is proved that shows that the spherical normal distribution mini- 1 mizes the trace of the Fisher information matrix given an entropy con- e-e ZH(X’+Y’) (5) straint-just as a sphere minimizes the surface area given a volume 2ae constraint.
    [Show full text]
  • Inequalities in Information Theory a Brief Introduction
    Inequalities in Information Theory A Brief Introduction Xu Chen Department of Information Science School of Mathematics Peking University Mar.20, 2012 Part I Basic Concepts and Inequalities Xu Chen (IS, SMS, at PKU) Inequalities in Information Theory Mar.20, 2012 2 / 80 Basic Concepts Outline 1 Basic Concepts 2 Basic inequalities 3 Bounds on Entropy Xu Chen (IS, SMS, at PKU) Inequalities in Information Theory Mar.20, 2012 3 / 80 Basic Concepts The Entropy Definition 1 The Shannon information content of an outcome x is defined to be 1 h(x) = log 2 P(x) 2 The entropy of an ensemble X is defined to be the average Shannon information content of an outcome: X 1 H(X ) = P(X ) log (1) 2 P(X ) x2X 3 Conditional Entropy: the entropy of a r.v.,given another r.v. X X H(X jY ) = − p(xi ; yj ) log2 p(xi jyj ) (2) i j Xu Chen (IS, SMS, at PKU) Inequalities in Information Theory Mar.20, 2012 4 / 80 Basic Concepts The Entropy Definition 1 The Shannon information content of an outcome x is defined to be 1 h(x) = log 2 P(x) 2 The entropy of an ensemble X is defined to be the average Shannon information content of an outcome: X 1 H(X ) = P(X ) log (1) 2 P(X ) x2X 3 Conditional Entropy: the entropy of a r.v.,given another r.v. X X H(X jY ) = − p(xi ; yj ) log2 p(xi jyj ) (2) i j Xu Chen (IS, SMS, at PKU) Inequalities in Information Theory Mar.20, 2012 4 / 80 Basic Concepts The Entropy Definition 1 The Shannon information content of an outcome x is defined to be 1 h(x) = log 2 P(x) 2 The entropy of an ensemble X is defined to be the average Shannon information content of an outcome: X 1 H(X ) = P(X ) log (1) 2 P(X ) x2X 3 Conditional Entropy: the entropy of a r.v.,given another r.v.
    [Show full text]
  • Quantitative Stability of the Entropy Power Inequality
    IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 64, NO. 8, AUGUST 2018 5691 Quantitative Stability of the Entropy Power Inequality Thomas A. Courtade , Member, IEEE,MaxFathi,andAshwinPananjady Abstract—We establish quantitative stability results for the these inequalities is reviewed in Section III-B. Due to the entropy power inequality (EPI). Specifically, we show that if formal equivalence of the Shannon-Stam and Lieb inequalities, uniformly log-concave densities nearly saturate the EPI, then they we shall generally refer to both as the EPI. must be close to Gaussian densities in the quadratic Kantorovich- Wasserstein distance. Furthermore, if one of the densities is It is well known that equality is achieved in the Shannon- Gaussian and the other is log-concave, or more generally has Stam EPI if and only if X and Y are Gaussian vectors with positive spectral gap, then the deficit in the EPI can be controlled proportional covariances. Equivalently, δEPI,t (µ, ν) vanishes if in terms of the L1-Kantorovich-Wasserstein distance or relative and only if µ, ν are Gaussian measures that are identical up entropy, respectively. As a counterpoint, an example shows to translation.1 However, despite the fundamental role the EPI that the EPI can be unstable with respect to the quadratic Kantorovich-Wasserstein distance when densities are uniformly plays in information theory and crisp characterization of equal- log-concave on sets of measure arbitrarily close to one. Our ity cases, few stability estimates are known. Specifically, our stability results can be extended to non-log-concave densities, motivating question is the following quantitative reinforcement provided certain regularity conditions are met.
    [Show full text]