On Concentration of Martingales and Applications in Information Theory, Communication & Coding
|
|
- Egbert Blair
- 5 years ago
- Views:
Transcription
1 On Concentration of Martingales and Applications in Information Theory, Communication & Coding Igal Sason Department of Electrical Engineering Technion - Israel Institute of Technology Haifa 32000, Israel ETH, Zurich, Switzerland August 22 23, I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
2 Concentration of Measures Concentration of Measures Concentration of measures is a central issue in probability theory, and it is strongly related to information theory and coding. Roughly speaking, the concentration of measure phenomenon can be stated in the following simple way: A random variable that depends in a smooth way on many independent random variables (but not too much on any of them) is essentially constant (Talagrand, 1996). I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
3 Concentration of Measures Concentration of Measures Concentration of measures is a central issue in probability theory, and it is strongly related to information theory and coding. Roughly speaking, the concentration of measure phenomenon can be stated in the following simple way: A random variable that depends in a smooth way on many independent random variables (but not too much on any of them) is essentially constant (Talagrand, 1996). Concentration Inequalities Concentration inequalities provide upper bounds on tail probabilities of the type P( X x t) (or P(X x t) for a random variable (RV) X, where x denotes the expectation or median of X. Several techniques were developed to prove concentration. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
4 Survey Papers on Concentration Inequalities for Martingales Survey Papers on Concentration Inequalities for Martingales 1 C. McDiarmid, Concentration, Probabilistic Methods for Algorithmic Discrete Mathematics, pp , Springer, F. Chung and L. Lu, Concentration inequalities and martingale inequalities: a survey, Internet Mathematics, vol. 3, no. 1, pp , March Available at fan/wp/concen.pdf. 3 I. S., On Refined Versions of the Azuma-Hoeffding Inequality with Applications in Information Theory, Communications and Coding, a tutorial paper with some original results, July Last updated in July See I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
5 Concentration via the Martingale Approach Definition - [Discrete-Time Martingales] Let (Ω, F, P) be a probability space. Let F 0 F 1... be a sequence of sub σ-algebras of F (which is called a filtration). A sequence X 0,X 1,... of RVs is a martingale if for every i 1 X i : Ω R is F i -measurable, i.e., 2 E[ X i ] <. {ω Ω : X i (ω) t} F i i {0,1,...},t R. 3 X i = E[X i+1 F i ] almost surely. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
6 Concentration via the Martingale Approach Definition - [Discrete-Time Martingales] Let (Ω, F, P) be a probability space. Let F 0 F 1... be a sequence of sub σ-algebras of F (which is called a filtration). A sequence X 0,X 1,... of RVs is a martingale if for every i 1 X i : Ω R is F i -measurable, i.e., 2 E[ X i ] <. {ω Ω : X i (ω) t} F i i {0,1,...},t R. 3 X i = E[X i+1 F i ] almost surely. A Simple Example: Random walk X n = n i=0 U i where P(U i = +1) = P(U i = 1) = 1 2 are i.i.d. RVs. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
7 Concentration via the Martingale Approach Martingales Remarks Remark 1 Given a RV X L 1 (Ω, F, P) and a filtration of sub σ-algebras {F i }, let X i = E[X F i ] i = 0,1,.... Then, the sequence X 0,X 1,... forms a martingale. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
8 Concentration via the Martingale Approach Martingales Remarks Remark 1 Given a RV X L 1 (Ω, F, P) and a filtration of sub σ-algebras {F i }, let X i = E[X F i ] i = 0,1,.... Then, the sequence X 0,X 1,... forms a martingale. Remark 2 Choose F 0 = {Ω, } and F n = F, then Remark 1 gives a martingale with X 0 = E[X F 0 ] = E[X] (F 0 doesn t provide information about X). X n = E[X F n ] = X a.s. (F n provides full information about X). I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
9 Concentration via the Martingale Approach Concentration via the Martingale Approach Theorem - [Azuma-Hoeffding inequality] Let {X k, F k } k=0 be a discrete-parameter real-valued martingale. If the jumps of the sequence {X k } are bounded almost surely (a.s.), i.e., X i X i 1 d i i = 1,2,...,n a.s. then ( P( X n X 0 r) 2exp r 2 2 n i=1 d2 i ), r > 0. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
10 Concentration via the Martingale Approach Concentration via the Martingale Approach Theorem - [Azuma-Hoeffding inequality] Let {X k, F k } k=0 be a discrete-parameter real-valued martingale. If the jumps of the sequence {X k } are bounded almost surely (a.s.), i.e., X i X i 1 d i i = 1,2,...,n a.s. then ( P( X n X 0 r) 2exp r 2 2 n i=1 d2 i ), r > 0. Concentration Inequalities Around the Average The Azuma-Hoeffding inequality and Remark 2 enable to get a concentration inequality for X around its expected value (E[X]). I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
11 Refined Versions of the Azuma-Hoeffding Inequality But, the Azuma-Hoeffding inequality is not tight!. For example, if r > n i=1 d i P( X n X 0 r) = 0. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
12 Refined Versions of the Azuma-Hoeffding Inequality Theorem (Th. 2 McDiarmid 89) Let {X k, F k } k=0 be a discrete-parameter real-valued martingale. Assume that, for some constants d,σ > 0, the following two requirements hold a.s. X k X k 1 d, Var(X k F k 1 ) = E [ (X k X k 1 ) 2 F k 1 ] σ 2 for every k {1,...,n}. Then, for every α 0, ( ( δ + γ )) γ P( X n X 0 αn) 2exp n D 1 + γ 1 + γ ( ) ( ) where γ σ2 d, δ α 2 d, and D(p q) p ln p q + (1 p)ln 1 p 1 q. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
13 Refined Versions of the Azuma-Hoeffding Inequality Corollary Under the conditions of Theorem 2, for every α 0, P( X n X 0 αn) 2exp ( nf(δ)) where f(δ) = { ln(2) [ 1 h 2 ( 1 δ 2 ) ], 0 δ 1 +, δ > 1 and h 2 (x) xlog 2 (x) (1 x)log 2 (1 x) for 0 x 1. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
14 Refined Versions of the Azuma-Hoeffding Inequality Corollary Under the conditions of Theorem 2, for every α 0, P( X n X 0 αn) 2exp ( nf(δ)) where f(δ) = { ln(2) [ 1 h 2 ( 1 δ 2 ) ], 0 δ 1 +, δ > 1 and h 2 (x) xlog 2 (x) (1 x)log 2 (1 x) for 0 x 1. Proof Set γ = 1 in Theorem 2. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
15 Refined Versions of the Azuma-Hoeffding Inequality Corollary Under the conditions of Theorem 2, for every α 0, P( X n X 0 αn) 2exp ( nf(δ)) where f(δ) = { ln(2) [ 1 h 2 ( 1 δ 2 ) ], 0 δ 1 +, δ > 1 and h 2 (x) xlog 2 (x) (1 x)log 2 (1 x) for 0 x 1. Proof Set γ = 1 in Theorem 2. Observation (first set γ = 1, and then use Pinsker s Inequality) Theorem 2 Corollary Azuma-Hoeffding inequality. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
16 Refined Versions of the Azuma-Hoeffding Inequality 2 LOWER BOUNDS ON EXPONENTS Theorem 2: γ=1/8 1/4 1/2 Corollary 2: f(δ) δ = α/d Exponent of Azuma inequality: δ 2 /2 I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
17 A Simple Example to Motivate Further Improvements A Simple Example to Motivate Further Improvements Let (Ω, F, P) be a probability space. d > 0 and γ (0,1] be fixed numbers. {U k } k N be i.i.d. random variables where P(U k = +d) = P(U k = d) = γ 2, P(U k = 0) = 1 γ, k N. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
18 A Simple Example to Motivate Further Improvements A Simple Example to Motivate Further Improvements Let (Ω, F, P) be a probability space. d > 0 and γ (0,1] be fixed numbers. {U k } k N be i.i.d. random variables where P(U k = +d) = P(U k = d) = γ 2, P(U k = 0) = 1 γ, k N. Consider the Markov process {X k } k 0 where X 0 = 0 and X k = X k 1 + U k, k N. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
19 A Simple Example to Motivate Further Improvements A Simple Example (Cont.) - Large Deviation Analysis From Cramér s theorem in R, for every α E[U 1 ] = 0, where the rate function is given by lim 1 n n ln P(X n αn) = I(α) I(α) = sup {tα ln E[exp(tU 1 )]}. t 0 I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
20 A Simple Example to Motivate Further Improvements A Simple Example (Cont.) - Large Deviation Analysis From Cramér s theorem in R, for every α E[U 1 ] = 0, where the rate function is given by lim 1 n n ln P(X n αn) = I(α) I(α) = sup {tα ln E[exp(tU 1 )]}. t 0 Let δ α d. Calculation shows that I(α) = δx ln (1 + γ [ cosh(x) 1 ]) ( δ(1 γ) + ) δ x ln 2 (1 γ) 2 + γ 2 (1 δ 2 ). γ(1 δ) I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
21 A Simple Example to Motivate Further Improvements A Simple Example (Cont.) - Large Deviation Analysis Lets examine the martingale approach in this simple case, where X k = with the natural filtration k U i, k N, X 0 = 0 i=1 F k = σ(u 1,...,U k ), k N, F 0 = {,Ω}. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
22 A Simple Example to Motivate Further Improvements A Simple Example (Cont.) - Large Deviation Analysis Lets examine the martingale approach in this simple case, where X k = with the natural filtration k U i, k N, X 0 = 0 i=1 F k = σ(u 1,...,U k ), k N, F 0 = {,Ω}. In this case, the martingale has bounded increments, and for every k N X k X k 1 d E[(X k X k 1 ) 2 F k 1 ] = Var(U k ) = γd 2 almost surely. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
23 A Simple Example to Motivate Further Improvements A Simple Example (Cont.) - Large Deviation Analysis Via the martingale approach, the following exponential inequality follows: ( ( δ + γ )) γ P (X n X 0 αn) exp n D 1 + γ 1 + γ where δ α d, and ( p ( 1 p ) D(p q) p ln + (1 p)ln, p,q [0,1] q) 1 q is the divergence (relative entropy) between the probability distributions (p,1 p) and (q,1 q). If δ > 1, then the above probability is zero. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
24 A Simple Example to Motivate Further Improvements A Simple Example (Cont.) - Large Deviation Analysis Via the martingale approach, the following exponential inequality follows: ( ( δ + γ )) γ P (X n X 0 αn) exp n D 1 + γ 1 + γ where δ α d, and ( p ( 1 p ) D(p q) p ln + (1 p)ln, p,q [0,1] q) 1 q is the divergence (relative entropy) between the probability distributions (p,1 p) and (q,1 q). If δ > 1, then the above probability is zero. This exponential bound is not tight (unless γ = 1), and the gap to the exact asymptotic exponent increases as the value of γ (0, 1] is decreased. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
25 A Simple Example to Motivate Further Improvements Comparison of the Exact Exponent and Its Lower Bound 0.8 COMPARISON BETWEEN THE EXPONENTS γ = 9/ δ = α/d I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
26 A Simple Example to Motivate Further Improvements Comparison of the Exact Exponent and Its Lower Bound (Cont.) 3 COMPARISON BETWEEN THE EXPONENTS γ = 1/ δ = α/d I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
27 A Simple Example to Motivate Further Improvements Questions: Why the lower bound on the exponent is not asymptotically tight? Why this gap increases as the value of γ is decreased? How this situation can be improved via the martingale approach? I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
28 A Simple Example to Motivate Further Improvements Questions: Why the lower bound on the exponent is not asymptotically tight? Why this gap increases as the value of γ is decreased? How this situation can be improved via the martingale approach? This exponential bound relies on Bennett s inequality: Since U k d, E[U k ] = 0, and Var(U k ) = γd 2, then γ exp(td) + exp( γtd) E [exp(tu k )], t γ I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
29 A Simple Example to Motivate Further Improvements Questions: Why the lower bound on the exponent is not asymptotically tight? Why this gap increases as the value of γ is decreased? How this situation can be improved via the martingale approach? This exponential bound relies on Bennett s inequality: Since U k d, E[U k ] = 0, and Var(U k ) = γd 2, then γ exp(td) + exp( γtd) E [exp(tu k )], t γ The probability distribution that achieves Bennett s inequality with equality is asymmetric (unless γ = 1), and is equal to P(Ũk = d) = γ 1 + γ, P(Ũk = γd) = γ. By reducing the value of γ (0,1], the above asymmetry grows. Indeed, this enlarges the gap to the exact asymptotic exponent. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
30 Conditionally Symmetric Martingales Interim Conclusion An improvement is likely to be obtained by a refinement of Bennett s inequality when X k is conditionally symmetric given F k 1. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
31 Conditionally Symmetric Martingales Interim Conclusion An improvement is likely to be obtained by a refinement of Bennett s inequality when X k is conditionally symmetric given F k 1. Definition: Conditionally Symmetric Martingales Let {X k, F k } k N0, where N 0 N {0}, be a discrete-time and real-valued martingale, and let ξ k X k X k 1 for every k N. Then {X k, F k } k N0 is a conditionally symmetric martingale if, conditioned on F k 1, the RV ξ k is symmetrically distributed around zero. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
32 Conditionally Symmetric Martingales Interim Conclusion An improvement is likely to be obtained by a refinement of Bennett s inequality when X k is conditionally symmetric given F k 1. Definition: Conditionally Symmetric Martingales Let {X k, F k } k N0, where N 0 N {0}, be a discrete-time and real-valued martingale, and let ξ k X k X k 1 for every k N. Then {X k, F k } k N0 is a conditionally symmetric martingale if, conditioned on F k 1, the RV ξ k is symmetrically distributed around zero. Definition: Conditionally Symmetric Sub/ Supermartingales Let {X k, F k } k N0 be a discrete-time real-valued sub or supermartingale, and let η k X k E[X k F k 1 ] for every k N. Then it is conditionally symmetric if, conditioned on F k 1, the RV η k is symmetrically distributed around zero. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
33 Conditionally Symmetric Martingales Construction of Conditionally Symmetric Martingales Example 1: Let (Ω, F, P) be a probability space {U k } k N L 1 (Ω, F, P) be independent random variables that are symmetrically distributed around zero {F k } k 0 be the natural filtration where F 0 = {,Ω} and F k = σ(u 1,...,U k ), k N. For k N, let A k L (Ω, F k 1, P) be an F k 1 -measurable random variable with a finite essential supremum. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
34 Conditionally Symmetric Martingales Construction of Conditionally Symmetric Martingales Example 1: Let (Ω, F, P) be a probability space {U k } k N L 1 (Ω, F, P) be independent random variables that are symmetrically distributed around zero {F k } k 0 be the natural filtration where F 0 = {,Ω} and F k = σ(u 1,...,U k ), k N. For k N, let A k L (Ω, F k 1, P) be an F k 1 -measurable random variable with a finite essential supremum. Define a new sequence of random variables in L 1 (Ω, F, P) where X n = n A k U k, n N k=1 and X 0 = 0. Then, {X n, F n } n N0 is a conditionally symmetric martingale. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
35 Conditionally Symmetric Martingales Construction of Conditionally Symmetric Martingales Example 2: Let {X n, F n } n N0 be a conditionally symmetric martingale A k L (Ω, F k 1, P) be an F k 1 -measurable random variable with a finite essential supremum. Define Y 0 = 0 and Y n = n k=1 A k(x k X k 1 ), n N. Then, {Y n, F n } n N0 is a conditionally symmetric martingale. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
36 Conditionally Symmetric Martingales Construction of Conditionally Symmetric Martingales Example 2: Let {X n, F n } n N0 be a conditionally symmetric martingale A k L (Ω, F k 1, P) be an F k 1 -measurable random variable with a finite essential supremum. Define Y 0 = 0 and Y n = n k=1 A k(x k X k 1 ), n N. Then, {Y n, F n } n N0 is a conditionally symmetric martingale. Example 3: A sampled Brownian motion is a discrete time, conditionally symmetric martingale with un-bounded increments. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
37 Conditionally Symmetric Martingales Construction of Conditionally Symmetric Martingales Example 2: Let {X n, F n } n N0 be a conditionally symmetric martingale A k L (Ω, F k 1, P) be an F k 1 -measurable random variable with a finite essential supremum. Define Y 0 = 0 and Y n = n k=1 A k(x k X k 1 ), n N. Then, {Y n, F n } n N0 is a conditionally symmetric martingale. Example 3: A sampled Brownian motion is a discrete time, conditionally symmetric martingale with un-bounded increments. Goal: Our next goal is to demonstrate how the assumption of the conditional symmetry improves existing exponential inequalities for discrete-time real-valued martingales with bounded increments. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
38 Exponential Inequalities for Conditionally Symmetric Martingales Exponential Inequalities for Conditionally Symmetric Martingales Lemma Let X be a real-valued RV with a symmetric distribution around zero, a support [ d,d], and assume Var(X) γd 2 for some d > 0 and γ [0,1]. Let h be a real-valued convex function, and assume that h(d 2 ) h(0). Then, E[h(X 2 )] (1 γ)h(0) + γh(d 2 ) with equality if P(X = ±d) = γ 2, P(X = 0) = 1 γ. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
39 Exponential Inequalities for Conditionally Symmetric Martingales Exponential Inequalities for Conditionally Symmetric Martingales Lemma Let X be a real-valued RV with a symmetric distribution around zero, a support [ d,d], and assume Var(X) γd 2 for some d > 0 and γ [0,1]. Let h be a real-valued convex function, and assume that h(d 2 ) h(0). Then, E[h(X 2 )] (1 γ)h(0) + γh(d 2 ) with equality if P(X = ±d) = γ 2, P(X = 0) = 1 γ. Proof h convex, X [ d,d] a.s. h(x 2 ) h(0) + ( ) X 2 ( d h(d 2 ) h(0) ). Taking expectations on both sides gives the inequality, with an equality for the above symmetric distribution. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
40 Exponential Inequalities for Conditionally Symmetric Martingales Exp. Inequalities for Conditionally Symmetric Martingales (Cont.) Corollary Let X be a real-valued RV with a symmetric distribution around zero, a support [ d,d], and assume Var(X) γd 2 for some d > 0 and γ [0,1]. Then, E [ exp(λx) ] 1 + γ [ cosh(λd) 1 ], λ R with equality if P(X = ±d) = γ 2, P(X = 0) = 1 γ. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
41 Exponential Inequalities for Conditionally Symmetric Martingales Exp. Inequalities for Conditionally Symmetric Martingales (Cont.) Corollary Let X be a real-valued RV with a symmetric distribution around zero, a support [ d,d], and assume Var(X) γd 2 for some d > 0 and γ [0,1]. Then, E [ exp(λx) ] 1 + γ [ cosh(λd) 1 ], λ R with equality if P(X = ±d) = γ 2, P(X = 0) = 1 γ. Proof Symmetric distribution of X E [ exp(λx) ] = E [ cosh(λx) ]. The corollary follows from the lemma since, for every x R, cosh(λx) = h(x 2 ) where h(x) λ 2n x n n=0 (2n)! is a convex function, and h(d 2 ) = cosh(λd) 1 = h(0). I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
42 Exponential Inequalities for Conditionally Symmetric Martingales Theorem 2 (I. S., 2012) Let {X k, F k } k N0 be a discrete-time real-valued and conditionally symmetric martingale. Assume that, for some fixed numbers d,σ > 0, the following two requirements are satisfied a.s. X k X k 1 d, Var(X k F k 1 ) = E [ (X k X k 1 ) 2 F k 1 ] σ 2 for every k N. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
43 Exponential Inequalities for Conditionally Symmetric Martingales Theorem 2 (I. S., 2012) Let {X k, F k } k N0 be a discrete-time real-valued and conditionally symmetric martingale. Assume that, for some fixed numbers d,σ > 0, the following two requirements are satisfied a.s. X k X k 1 d, Var(X k F k 1 ) = E [ (X k X k 1 ) 2 F k 1 ] σ 2 for every k N. Then, for every α 0 and n N, ( ) P max X k X 0 αn 2exp ( ne(γ,δ) ) 1 k n where γ σ2 d 2, δ α d and for γ (0,1] and δ [0,1), the exponent E(γ,δ) is given as follows: I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
44 Exponential Inequalities for Conditionally Symmetric Martingales Theorem 2 (Cont.) E(γ,δ) δx ln (1 + γ [ cosh(x) 1 ]) ( δ(1 γ) + ) δ x ln 2 (1 γ) 2 + γ 2 (1 δ 2 ). γ(1 δ) If δ > 1, then the probability is zero (so E(γ,δ) + ), and E(γ,1) = ln ( 2 γ). I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
45 Exponential Inequalities for Conditionally Symmetric Martingales Theorem 2 (Cont.) E(γ,δ) δx ln (1 + γ [ cosh(x) 1 ]) ( δ(1 γ) + ) δ x ln 2 (1 γ) 2 + γ 2 (1 δ 2 ). γ(1 δ) If δ > 1, then the probability is zero (so E(γ,δ) + ), and E(γ,1) = ln ( 2 γ). Asymptotic Optimality of the Exponent The example we studied earlier shows that the exponent of the bound in Theorem 1 is asymptotically optimal. That is, there exists a conditionally symmetric martingale, satisfying the conditions in this theorem, that attains the exponent E(γ,δ) in the limit where n. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
46 Exponential Inequalities for Conditionally Symmetric Martingales Theorem 2 should be compared to the bound in Theorem 1 which does not require the conditional symmetry property. Theorem 1 (Reminder) [McDiarmid 1989, book of Dembo & Zeitouni (Corollary 2.4.7)] Let {X k, F k } k N0 be a discrete-time real-valued martingale with bounded jumps. Assume that the two conditions on the bounded increments and conditional variance from Theorem 1 are satisfied a.s. for every k N. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
47 Exponential Inequalities for Conditionally Symmetric Martingales Theorem 2 should be compared to the bound in Theorem 1 which does not require the conditional symmetry property. Theorem 1 (Reminder) [McDiarmid 1989, book of Dembo & Zeitouni (Corollary 2.4.7)] Let {X k, F k } k N0 be a discrete-time real-valued martingale with bounded jumps. Assume that the two conditions on the bounded increments and conditional variance from Theorem 1 are satisfied a.s. for every k N. Then, for every α 0 and n N, ( ) P max X k X 0 αn 1 k n ( 2exp n D ( δ + γ )) γ 1 + γ 1 + γ where γ σ2 d and δ α 2 d were introduced earlier, and ( p ( 1 p ) D(p q) p ln + (1 p)ln, p,q [0,1]. q) 1 q If δ > 1, then the probability is zero. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
48 Exponential Inequalities for Conditionally Symmetric Martingales Proof Technique for Theorems 1 and 2 Define X n X 0 = n k=1 ξ k where ξ k X k X k 1. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
49 Exponential Inequalities for Conditionally Symmetric Martingales Proof Technique for Theorems 1 and 2 Define X n X 0 = n k=1 ξ k where ξ k X k X k 1. ξ k d, E [ ξ k F k 1 ] = 0, Var(ξk F k 1 ) γd 2. The RV ξ k is F k -measurable. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
50 Exponential Inequalities for Conditionally Symmetric Martingales Proof Technique for Theorems 1 and 2 Define X n X 0 = n k=1 ξ k where ξ k X k X k 1. ξ k d, E [ ξ k F k 1 ] = 0, Var(ξk F k 1 ) γd 2. The RV ξ k is F k -measurable. Exponentiation and the maximal inequality for submartingales give that, for every α 0, ( ) [ ( P max (X k X 0 ) nα e nαt E exp t 1 k n n ξ k )], t 0. k=1 I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
51 Exponential Inequalities for Conditionally Symmetric Martingales Proof Technique for Theorems 1 and 2 Define X n X 0 = n k=1 ξ k where ξ k X k X k 1. ξ k d, E [ ξ k F k 1 ] = 0, Var(ξk F k 1 ) γd 2. The RV ξ k is F k -measurable. Exponentiation and the maximal inequality for submartingales give that, for every α 0, ( ) [ ( P max (X k X 0 ) nα e nαt E exp t 1 k n n ξ k )], t 0. k=1 Since {F i } forms a filtration, then for every t 0 [ ( n )] [ ( n 1 ) E exp t ξ k = E exp t ξ k E [ ] ] exp(tξ n ) F n 1. k=1 k=1 I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
52 Exponential Inequalities for Conditionally Symmetric Martingales Proof Technique for Theorems 1 and 2 (Cont.) For proving Theorem 2, the use of Bennett s inequality gives E [exp(tξ k ) F k 1 ] γ exp(td) + exp( γtd). 1 + γ I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
53 Exponential Inequalities for Conditionally Symmetric Martingales Proof Technique for Theorems 1 and 2 (Cont.) For proving Theorem 2, the use of Bennett s inequality gives E [exp(tξ k ) F k 1 ] γ exp(td) + exp( γtd). 1 + γ For the refinement in Theorem 1 for conditionally symmetric martingales with bounded increments, use of Lemma 1 gives E [exp(tξ k ) F k 1 ] 1 + γ [ cosh(td) 1 ]. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
54 Exponential Inequalities for Conditionally Symmetric Martingales Proof Technique for Theorems 1 and 2 (Cont.) For proving Theorem 2, the use of Bennett s inequality gives E [exp(tξ k ) F k 1 ] γ exp(td) + exp( γtd). 1 + γ For the refinement in Theorem 1 for conditionally symmetric martingales with bounded increments, use of Lemma 1 gives E [exp(tξ k ) F k 1 ] 1 + γ [ cosh(td) 1 ]. In both cases, one gets exponential bounds with the free parameter t. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
55 Exponential Inequalities for Conditionally Symmetric Martingales Proof Technique for Theorems 1 and 2 (Cont.) For proving Theorem 2, the use of Bennett s inequality gives E [exp(tξ k ) F k 1 ] γ exp(td) + exp( γtd). 1 + γ For the refinement in Theorem 1 for conditionally symmetric martingales with bounded increments, use of Lemma 1 gives E [exp(tξ k ) F k 1 ] 1 + γ [ cosh(td) 1 ]. In both cases, one gets exponential bounds with the free parameter t. Optimization over t 0 & union bound to get two-sided inequalities. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
56 Exponential Inequalities for Conditionally Symmetric Martingales Proof Technique for Theorems 1 and 2 (Cont.) For proving Theorem 2, the use of Bennett s inequality gives E [exp(tξ k ) F k 1 ] γ exp(td) + exp( γtd). 1 + γ For the refinement in Theorem 1 for conditionally symmetric martingales with bounded increments, use of Lemma 1 gives E [exp(tξ k ) F k 1 ] 1 + γ [ cosh(td) 1 ]. In both cases, one gets exponential bounds with the free parameter t. Optimization over t 0 & union bound to get two-sided inequalities. The asymptotic optimality of the exponent in Theorem 1 follows from the simple example that was shown earlier. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
57 Relation to Classical Results in Probability Theory Relation to Classical Results in Probability Theory: The above concentration inequalities are linked to Central limit theorem for martingales Law of iterated logarithm Moderate deviations principle for real-valued i.i.d. RVs. For proofs and discussions on these relations, see Section IV in I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
58 Application 1: MDP for Binary Hypothesis Testing Binary Hypothesis Testing Let the RVs X 1,X 2... be i.i.d. Q, and consider two hypotheses: H 1 : Q = P 1. H 2 : Q = P 2. For simplicity, assume that the RVs are discrete, and take their values on a finite alphabet X where P 1 (x),p 2 (x) > 0 for every x X. The log-likelihood ratio (LLR): L(X n 1 ) = n i=1 ln P 1(X i ) P 2 (X i ). By the strong law of large numbers (SLLN), if H 1 is true, then a.s. L(X1 lim n) L(X1 n n = D(P 1 P 2 ) and, if H 2, lim n) n n = D(P 2 P 1 ). Let λ,λ R satisfy D(P 2 P 1 ) < λ λ < D(P 1 P 2 ). Decide on H 1 if L(X n 1 ) > nλ and on H 2 if L(X n 1 ) < nλ. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
59 Application 1: MDP for Binary Hypothesis Testing Let and then ) α (1) n P 1 (L(X n 1 n ) nλ ( ) β n (1) P2 n L(X1 n ) nλ ) α (2) n P 1 (L(X n 1 n ) nλ ( ) β n (2) P2 n L(X1 n ) nλ α (1) n and β n (1) are the probabilities of either making an error/ declaring an erasure under, respectively, hypotheses H 1 and H 2. α (2) n and β n (2) are the probabilities of making an error under, respectively, hypotheses H 1 and H 2. Large deviations analysis α n and β n both decay exponentially to zero as a function of the block length (n). I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
60 Application 1: MDP for Binary Hypothesis Testing Moderate-Deviations Analysis for Binary Hypothesis Testing Instead of the setting where the thresholds are kept fixed independently of n, let these thresholds tend to their asymptotic limits (due to the SLLN), i.e., lim n λ(n) = D(P 1 P 2 ), lim n λ(n) = D(P 2 P 1 ). In moderate-deviations analysis of binary hypothesis testing, we are interested to analyze the case where 1 The block length n of the input sequence tends to infinity. 2 The thresholds tend to their asymptotic limits simultaneously. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
61 Application 1: MDP for Binary Hypothesis Testing Moderate-Deviations Analysis for Binary Hypothesis Testing (Cont.) To this end, let η ( 1 2,1), and ε 1,ε 2 > 0 be arbitrary fixed numbers. Set the upper and lower thresholds to λ (n) = D(P 1 P 2 ) ε 1 n (1 η) λ (n) = D(P 2 P 1 ) + ε 2 n (1 η). I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
62 Application 1: MDP for Binary Hypothesis Testing Moderate-Deviations Analysis via the Martingale Approach Under hypothesis H 1, construct the martingale {U k, F k } n k=0 where F 0 F 1...F n is the natural filtration F 0 = {,Ω}, F k = σ(x 1,...,X k ), k {1,...,n}. U k = E P n 1 [ L(X n 1 ) F k ]. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
63 Application 1: MDP for Binary Hypothesis Testing Moderate-Deviations Analysis via the Martingale Approach Under hypothesis H 1, construct the martingale {U k, F k } n k=0 where F 0 F 1...F n is the natural filtration F 0 = {,Ω}, F k = σ(x 1,...,X k ), k {1,...,n}. U k = E P n 1 [ L(X n 1 ) F k ]. For every k {0,...,n}: U k = k i=1 ln P 1(X i ) P 2 (X i ) + (n k)d(p 1 P 2 ). I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
64 Application 1: MDP for Binary Hypothesis Testing Moderate-Deviations Analysis via the Martingale Approach (Cont.) U 0 = nd(p 1 P 2 ), and U n = L(X n 1 ). Let d 1 max x X ln P 1(x) P 2 (x) D(P 1 P 2 ) d 1 <. U k U k 1 d 1 holds a.s. for every k {1,...,n}. Due to the independence of the RVs in the sequence {X i } [ E P n (Uk 1 U k 1 ) 2 ] F k 1 = { ( P 1 (x) ln P ) } 2 1(x) P 2 (x) D(P 1 P 2 ) x X σ 2 1. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
65 Application 1: MDP for Binary Hypothesis Testing Moderate-Deviations Analysis via the Martingale Approach (Cont.) Let ε 1 > 0 and η ( 1 2,1) be two arbitrarily fixed numbers as above. Under hypothesis H 1, it follows from Theorem 2 and the above construction of a martingale that ( ( P1 n ( L(X n 1 ) nλ (n) (η,n) ) δ 1 + γ 1 ) ) exp nd γ γ γ 1 where with d 1 and σ 2 1 δ (η,n) 1 ε 1n (1 η) d 1, γ 1 σ2 1 d 2 1 as defined in the previous slide. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
66 Application 1: MDP for Binary Hypothesis Testing Moderate-Deviations Analysis via the Martingale Approach (Cont.) The concentration inequality in Theorem 2 and ( a simple ) inequality give that, under hypothesis H 1, for every η 1 2,1, α (1) n ( exp ( ε2 1 n2η 1 2σ1 2 1 ε 1 d 1 3σ 2 1 (1 + γ 1) )) 1 n 1 η so this upper bound on the overall probability of either making an error or declaring an erasure under hypothesis H 1 decays sub-exponentially to zero. It improves by increasing η ( 1 2,1). On the other hand, the exponential decay of the probability of error improves as the value of η ( 1 2,1) is decreased (since the margin for making an error is increased). α (2) n I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
67 Application 1: MDP for Binary Hypothesis Testing Moderate-Deviations Analysis via the Martingale Approach (Cont.) Moderate-deviations analysis reflects, as can be expected, a tradeoff between the two α n s and also between the two β n s. But all decay asymptotically to zero as n tends to infinity (which is indeed consistent with the SLLN). I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
68 Application 1: MDP for Binary Hypothesis Testing Moderate-Deviations Analysis via the Martingale Approach (Cont.) The following upper bound implies that lim n n1 2η ln P1 n ( L(X n 1 ) nλ (n) ) ε 2 1 2σ1 2. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
69 Application 1: MDP for Binary Hypothesis Testing Moderate-Deviations Analysis via the Martingale Approach (Cont.) The following upper bound implies that lim n n1 2η ln P1 n ( L(X n 1 ) nλ (n) ) ε 2 1 2σ1 2. Question: But, does the upper bound reflect the correct asymptotic scaling of this probability? I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
70 Application 1: MDP for Binary Hypothesis Testing Moderate-Deviations Analysis via the Martingale Approach (Cont.) The following upper bound implies that lim n n1 2η ln P1 n ( L(X n 1 ) nλ (n) ) ε 2 1 2σ1 2. Question: But, does the upper bound reflect the correct asymptotic scaling of this probability? Reply: Indeed, it follows from the moderate-deviations principle (MDP) for real-valued RVs. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
71 Application 1: MDP for Binary Hypothesis Testing Moderate-Deviations Principle (MDP) for Real-Valued RVs Theorem Let {X i } be i.i.d. random real-valued RVs, satisfying E[X i ] = 0, Var(X i ) = σ 2, X i d and let η ( 1 2,1). Then, for every α > 0, ( ) n lim n n1 2η ln P X i αn η i=1 = α2 2σ2, α 0. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
72 Application 1: MDP for Binary Hypothesis Testing Moderate-Deviations Analysis via the Martingale Approach (Cont.) The MDP shows that this inequality holds in fact with equality. The refined inequality in Theorem 2 gives the correct asymptotic scaling in this case. It also gives an analytical bound for finite n. This is in contrast to the analysis which follows from the Azuma-Hoeffding inequality, which does not coincide with the correct asymptotic scaling (since ε2 1 2σ 2 1 is replaced by ε2 1, and σ 2d 2 1 d 1 ). 1 In the considered setting of moderate-deviations analysis for binary hypothesis testing, the error probability decays sub-exponentially to 0. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
73 Application 1: MDP for Binary Hypothesis Testing Papers on Moderate-Deviations Analysis in IT Aspects 1 E. A. Abbe, Local to Global Geometric Methods in Information Theory, Ph.D. dissertation, MIT, Boston, MA, USA, June Y. Altuǧ and A. B. Wagner, Moderate-deviations analysis of channel coding: discrete memoryless case, Proc. ISIT 2010, pp , Austin, Texas, USA, June D. He, L. A. Lastras-Montaño, E. Yang, A. Jagmohan and J. Chen, On the redundancy of Slepian-Wolf coding, IEEE Trans. on Information Theory, vol. 55, no. 12, pp , December Y. Polyanskiy and S. Verdú, Channel dispersion and moderate-deviations limits of memoryless channels, Proceedings Forty-Eighth Annual Allerton Conference, pp , UIUC, Illinois, USA, October I. Sason, Moderate-deviations analysis of binary hypothesis testing, November V. Y. F. Tan, Moderate-deviations of lossy source coding for discrete and Gaussian channels, November I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
74 Application 2 - Concentration of Martingales in Coding Theory Concentration Phenomena for Codes Defined on Graphs Motivation & Background The performance analysis of a particular code is difficult, especially for codes of large block lengths. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
75 Application 2 - Concentration of Martingales in Coding Theory Concentration Phenomena for Codes Defined on Graphs Motivation & Background The performance analysis of a particular code is difficult, especially for codes of large block lengths. Does the performance concentrate around the average performance of the ensemble? I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
76 Application 2 - Concentration of Martingales in Coding Theory Concentration Phenomena for Codes Defined on Graphs Motivation & Background The performance analysis of a particular code is difficult, especially for codes of large block lengths. Does the performance concentrate around the average performance of the ensemble? The existence of such a concentration validates the use of the density evolution technique as an analytical tool to assess performance of long enough codes (e.g., LDPC codes) and to assess their asymptotic gap to capacity. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
77 Application 2 - Concentration of Martingales in Coding Theory Concentration Phenomena for Codes Defined on Graphs Motivation & Background The performance analysis of a particular code is difficult, especially for codes of large block lengths. Does the performance concentrate around the average performance of the ensemble? The existence of such a concentration validates the use of the density evolution technique as an analytical tool to assess performance of long enough codes (e.g., LDPC codes) and to assess their asymptotic gap to capacity. The current concentration results for codes defined on graphs, which mainly rely on the Azuma-Hoeffding inequality, are weak since in practice concentration is observed at much shorter block lengths. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
78 Application 2 - Concentration of Martingales in Coding Theory Performance under Message-Passing Decoding Theorem I - [Concentration of performance under iterative message-passing decoding (Richardson and Urbanke, 2001)] Let C, a code chosen uniformly at random from the ensemble LDPC(n,λ,ρ), be used for transmission over a memoryless binary-input output-symmetric (MBIOS) channel. Assume that the decoder performs l iterations of message-passing decoding, and let P b (C,l) denote the resulting bit error probability. Then, for every δ > 0, there exists an α > 0 where α = α(λ,ρ,δ,l) (independent of the block length n) such that P ( P b (C,l) E LDPC(n,λ,ρ) [P b (C,l)] δ ) e αn Proof The proof applies Azuma s inequality to a martingale sequence with bounded differences (IEEE Trans. on IT, Feb. 2001). I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
79 Application 2 - Concentration of Martingales in Coding Theory Conditional Entropy of LDPC code ensembles Theorem II - [Concentration of Conditional Entropy of LDPC code ensembles (Méasson et al. 2008)] Let C be chosen uniformly at random from the ensemble LDPC(n,λ,ρ). Assume that the transmission of the code C takes place over an MBIOS channel. Let H(X Y) designate the conditional entropy of the transmitted codeword X given the received sequence Y from the channel. Then, for any ξ > 0, P ( H(X Y) E LDPC(n,λ,ρ) [H(X Y)] n ξ ) 2exp( Bξ 2 ) 1 where B 2(d max c +1) 2 (1 R d ), dmax c is the maximal check-node degree, and R d is the design rate of the ensemble. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
80 Application 2 - Concentration of Martingales in Coding Theory Proof - [outline] 1 Introduction of a martingale sequence with bounded differences: Define the RV Z = HG (X Y), where G is a graph of a code chosen uniformly at random from the ensemble LDPC(n, λ, ρ) Define the martingale sequence Zt = E[Z F t ] t {0, 1,..., m}, where the filtration is the sequence of subsets of σ-algebras, generated by revealing each time another parity-check equation of the code. 2 Upper bounds on the differences Z t+1 Z t : It was proved that Zt+1 Z t (r + 1)H G ( X Y), where r is the degree of parity-check equation revealed at time t, and X = X i1... X ir (i.e., X is the modulo-2 sum of some r bits in the codeword X). Then r d max c, and H G ( X Y) 1. 3 Azuma s inequality was applied to a get a concentration inequality, using Z t+1 Z t d max c + 1 for every t = 0,...,m 1 where m = n(1 R d ) is the number of parity-check nodes. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
81 Application 2 - Concentration of Martingales in Coding Theory Improvements to Theorem II Improvement 1 - A tightened upper bound on the conditional entropy Instead of upper bounding H G ( X Y) by 1, which is independent of the channel capacity (C), it can be proved that ( ) H G ( X Y) 1 C r 2 h 2 where h is the binary entropy function to the base 2. For a BSC or BEC, this bound can be further improved. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
82 Application 2 - Concentration of Martingales in Coding Theory Improvement 2 (trivial) Instead of taking the trivial bound r d max c for all m terms in the Azuma s inequality, one can rely on the degree distribution of the parity-check nodes. The number of parity-check nodes of degree r is n(1 R d )Γ r. Theorem III - [Tightened Expressions for B] Considering the terms of Theorem II, applying these two improvements yields tightened expressions for B. 1 General MBIOS - B [ ( BSC - B 1 2(1 R d ) d max c i=1 (i+1)2 Γ i h 1 C i 2 2 2(1 R d ) d max ( )] c 1 [1 2h i=1 (i+1)2 1 (1 C)] i 2 Γ i [h 2 BEC - B 1 2(1 R d ) d max c i=1 (i+1)2 Γ i (1 C i ) 2 )] 2 I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
83 Application 2 - Concentration of Martingales in Coding Theory Numerical comparison for BEC and BIAWGN Let us consider the case where (2,20) regular LDPC code ensemble. Communication over a BEC or BIAWGNC with capacity of 0.98 per channel use. Compared to Theorems II, applying Theorem III results in tighter expressions for B: [ ( )] 2 BIAWGN - Improvement by factor h = C dc 2 2 BEC - Improvement by factor 1 (1 C dc ) 2 = I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
84 Application 2 - Concentration of Martingales in Coding Theory Comparison for Heavy-Tail Poisson Distribution (Tornado Codes) Consider the capacity-achieving Tornado LDPC code ensemble for a BEC with erasure probability p. We wish to design a code ensemble that achieves a fraction 1 ε of the capacity. Theorem II - The parity-check degree is Poisson distributed, therefore =. Hence, B = 0 and this result is useless. ( ) 1 Theorem III - B scales (at least) like O ). d max c log 2( 1 ε The parameter B tends to zero slowly as we let the fractional gap ε tend to zero; this demonstrates a rather fast concentration. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
85 Application 2 - Concentration of Martingales in Coding Theory Interim Conclusions The use of Azuma s inequality was addressed in the context of proving concentration phenomena for code ensembles defined on graphs and iterative decoding algorithms. A possible tightening of a concentration inequality for the conditional entropy of LDPC ensembles by using Azuma s inequality, and deriving an improved upper bound on the jumps of the martingale sequence. The improved inequality enables to prove concentration of the conditional entropy for ensembles of Tornado codes (in contrast to the original concentration inequality). I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
86 Application 2 - Concentration of Martingales in Coding Theory Concentration of Martingales in Coding Theory Selected Papers Applying Azuma s Inequality for LDPC Ensembles M. Sipser and D. A. Spielman, Expander codes, IEEE Trans. on Information Theory, vol. 42, no. 6, pp , November M.G. Luby, M. Mitzenmacher, M.A. Shokrollahi and D.A. Spielman, Efficient erasure correcting codes, IEEE Trans. on Information Theory, vol. 47, no. 2, pp , February T. Richardson and R. Urbanke, The capacity of low-density parity check codes under message-passing decoding. IEEE Trans. on Information Theory, vol. 47, pp , February A. Kavcic, X. Ma and M. Mitzenmacher, Binary intersymbol interference channels: Gallager bounds, density evolution, and code performance bounds, IEEE Trans. on Information Theory, vol. 49, no. 7, pp , July I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
87 Application 2 - Concentration of Martingales in Coding Theory (Cont.) A. Montanari, Tight bounds for LDPC and LDGM codes under MAP decoding, IEEE Trans. on Information Theory, vol. 51, no. 9, pp , September C. Méasson, A. Montanari and R. Urbanke, Maxwell construction: The hidden bridge between iterative and maximum a-posteriori decoding, IEEE Trans. on Information Theory, vol. 54, pp , December L.R. Varshney, Performance of LDPC codes under faulty iterative decoding, IEEE Trans. on Information Theory, vol. 57, no. 7, pp , July I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
88 Application 3: Concentration for OFDM Signals Orthogonal Frequency Division Multiplexing (OFDM) The OFDM modulation converts a high-rate data stream into a number of low-rate steams that are transmitted over parallel narrow-band channels. One of the problems of OFDM is that the peak amplitude of the signal can be significantly higher than the average amplitude. Sensitivity to non-linear devices in the communication path (e.g., digital-to-analog converters, mixers and high-power amplifiers). An increase in the symbol error rate and also a reduction in the power efficiency as compared to single-carrier systems. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
89 Application 3: Concentration for OFDM Signals OFDM (Cont.) Given an n-length codeword {X i } i=0 n 1, a single OFDM baseband symbol is described by s(t;x 0,...,X n 1 ) = 1 n 1 ( j 2πit ) X i exp, 0 t T. n T i=0 Assume that X 0,...,X n 1 are i.i.d. complex RVs with X i = 1. Since the sub-carriers are orthonormal over [0,T], then a.s. the power of the signal s over this interval is 1. The CF of the signal s, composed of n sub-carriers, is defined as CF n (s) max 0 t T s(t). The CF scales with high probability like log(n) for large n. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
90 Application 3: Concentration for OFDM Signals Concentration of Measures In the following, we consider two of the main approaches for proving concentration inequalities, and apply them to prove concentration for the crest factor of OFDM signals. 1 The 1st approach is based on martingales (the Azuma-Hoeffding inequality and some refinements). 2 The 2nd approach is based on Talagrand s inequalities. I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
91 Application 3: Concentration for OFDM Signals A Previously Reported Result A concentration inequality for the CF of OFDM signals was derived (Litsyn and Wunder, IEEE Trans. on IT, 2006). It states that for every c 2.5 CFn P( (s) ) ( clog log(n) 1 log(n) < = 1 O ( ) log(n) 4 ). log(n) I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
92 Application 3: Concentration for OFDM Signals Theorem - [McDiarmid s Inequality] Let X = (X 1,...,X n ) be a vector of independent random variables with X k taking values in a set A k for each k. Suppose that a real-valued function f, defined on k A k, satisfies f(x) f(x ) c k whenever the vectors x and x differ only in the k-th coordinate. Let µ E[f(X)] be the expected value of f(x). Then, for every α 0, P( f(x) µ α) 2exp ) ( 2α2. k c2 k I. Sason (Technion) Seminar Talk, ETH, Zurich, Switzerland August 22 23, / 90
On the Concentration of the Crest Factor for OFDM Signals
On the Concentration of the Crest Factor for OFDM Signals Igal Sason Department of Electrical Engineering Technion - Israel Institute of Technology Haifa 32000, Israel The 2011 IEEE International Symposium
More informationOn the Concentration of the Crest Factor for OFDM Signals
On the Concentration of the Crest Factor for OFDM Signals Igal Sason Department of Electrical Engineering Technion - Israel Institute of Technology, Haifa 3, Israel E-mail: sason@eetechnionacil Abstract
More informationLower Bounds on the Graphical Complexity of Finite-Length LDPC Codes
Lower Bounds on the Graphical Complexity of Finite-Length LDPC Codes Igal Sason Department of Electrical Engineering Technion - Israel Institute of Technology Haifa 32000, Israel 2009 IEEE International
More informationConcentration of Measure with Applications in Information Theory, Communications and Coding
Concentration of Measure with Applications in Information Theory, Communications and Coding Maxim Raginsky UIUC Urbana, IL 61801, USA maxim@illinois.edu Igal Sason Technion Haifa 32000, Israel sason@ee.technion.ac.il
More informationOn the Entropy of Sums of Bernoulli Random Variables via the Chen-Stein Method
On the Entropy of Sums of Bernoulli Random Variables via the Chen-Stein Method Igal Sason Department of Electrical Engineering Technion - Israel Institute of Technology Haifa 32000, Israel ETH, Zurich,
More informationOn Generalized EXIT Charts of LDPC Code Ensembles over Binary-Input Output-Symmetric Memoryless Channels
2012 IEEE International Symposium on Information Theory Proceedings On Generalied EXIT Charts of LDPC Code Ensembles over Binary-Input Output-Symmetric Memoryless Channels H Mamani 1, H Saeedi 1, A Eslami
More informationEE229B - Final Project. Capacity-Approaching Low-Density Parity-Check Codes
EE229B - Final Project Capacity-Approaching Low-Density Parity-Check Codes Pierre Garrigues EECS department, UC Berkeley garrigue@eecs.berkeley.edu May 13, 2005 Abstract The class of low-density parity-check
More informationPerformance Analysis and Code Optimization of Low Density Parity-Check Codes on Rayleigh Fading Channels
Performance Analysis and Code Optimization of Low Density Parity-Check Codes on Rayleigh Fading Channels Jilei Hou, Paul H. Siegel and Laurence B. Milstein Department of Electrical and Computer Engineering
More informationOn Achievable Rates and Complexity of LDPC Codes over Parallel Channels: Bounds and Applications
On Achievable Rates and Complexity of LDPC Codes over Parallel Channels: Bounds and Applications Igal Sason, Member and Gil Wiechman, Graduate Student Member Abstract A variety of communication scenarios
More informationCapacity-Achieving Ensembles for the Binary Erasure Channel With Bounded Complexity
Capacity-Achieving Ensembles for the Binary Erasure Channel With Bounded Complexity Henry D. Pfister, Member, Igal Sason, Member, and Rüdiger Urbanke Abstract We present two sequences of ensembles of non-systematic
More informationAnalytical Bounds on Maximum-Likelihood Decoded Linear Codes: An Overview
Analytical Bounds on Maximum-Likelihood Decoded Linear Codes: An Overview Igal Sason Department of Electrical Engineering, Technion Haifa 32000, Israel Sason@ee.technion.ac.il December 21, 2004 Background
More informationMarch 1, Florida State University. Concentration Inequalities: Martingale. Approach and Entropy Method. Lizhe Sun and Boning Yang.
Florida State University March 1, 2018 Framework 1. (Lizhe) Basic inequalities Chernoff bounding Review for STA 6448 2. (Lizhe) Discrete-time martingales inequalities via martingale approach 3. (Boning)
More informationBounds on Achievable Rates of LDPC Codes Used Over the Binary Erasure Channel
Bounds on Achievable Rates of LDPC Codes Used Over the Binary Erasure Channel Ohad Barak, David Burshtein and Meir Feder School of Electrical Engineering Tel-Aviv University Tel-Aviv 69978, Israel Abstract
More informationRecent Results on Capacity-Achieving Codes for the Erasure Channel with Bounded Complexity
26 IEEE 24th Convention of Electrical and Electronics Engineers in Israel Recent Results on Capacity-Achieving Codes for the Erasure Channel with Bounded Complexity Igal Sason Technion Israel Institute
More informationTightened Upper Bounds on the ML Decoding Error Probability of Binary Linear Block Codes and Applications
on the ML Decoding Error Probability of Binary Linear Block Codes and Department of Electrical Engineering Technion-Israel Institute of Technology An M.Sc. Thesis supervisor: Dr. Igal Sason March 30, 2006
More informationP (A G) dp G P (A G)
First homework assignment. Due at 12:15 on 22 September 2016. Homework 1. We roll two dices. X is the result of one of them and Z the sum of the results. Find E [X Z. Homework 2. Let X be a r.v.. Assume
More informationTightened Upper Bounds on the ML Decoding Error Probability of Binary Linear Block Codes and Applications
on the ML Decoding Error Probability of Binary Linear Block Codes and Moshe Twitto Department of Electrical Engineering Technion-Israel Institute of Technology Haifa 32000, Israel Joint work with Igal
More informationBounds on the Performance of Belief Propagation Decoding
Bounds on the Performance of Belief Propagation Decoding David Burshtein and Gadi Miller Dept. of Electrical Engineering Systems Tel-Aviv University Tel-Aviv 69978, Israel Email: burstyn@eng.tau.ac.il,
More informationOptimal Rate and Maximum Erasure Probability LDPC Codes in Binary Erasure Channel
Optimal Rate and Maximum Erasure Probability LDPC Codes in Binary Erasure Channel H. Tavakoli Electrical Engineering Department K.N. Toosi University of Technology, Tehran, Iran tavakoli@ee.kntu.ac.ir
More informationAn Improved Sphere-Packing Bound for Finite-Length Codes over Symmetric Memoryless Channels
An Improved Sphere-Packing Bound for Finite-Length Codes over Symmetric Memoryless Channels Gil Wiechman Igal Sason Department of Electrical Engineering Technion, Haifa 3000, Israel {igillw@tx,sason@ee}.technion.ac.il
More informationSlepian-Wolf Code Design via Source-Channel Correspondence
Slepian-Wolf Code Design via Source-Channel Correspondence Jun Chen University of Illinois at Urbana-Champaign Urbana, IL 61801, USA Email: junchen@ifpuiucedu Dake He IBM T J Watson Research Center Yorktown
More informationNetwork coding for multicast relation to compression and generalization of Slepian-Wolf
Network coding for multicast relation to compression and generalization of Slepian-Wolf 1 Overview Review of Slepian-Wolf Distributed network compression Error exponents Source-channel separation issues
More informationLow-density parity-check codes
Low-density parity-check codes From principles to practice Dr. Steve Weller steven.weller@newcastle.edu.au School of Electrical Engineering and Computer Science The University of Newcastle, Callaghan,
More informationAn Introduction to Low Density Parity Check (LDPC) Codes
An Introduction to Low Density Parity Check (LDPC) Codes Jian Sun jian@csee.wvu.edu Wireless Communication Research Laboratory Lane Dept. of Comp. Sci. and Elec. Engr. West Virginia University June 3,
More informationarxiv: v4 [cs.it] 17 Oct 2015
Upper Bounds on the Relative Entropy and Rényi Divergence as a Function of Total Variation Distance for Finite Alphabets Igal Sason Department of Electrical Engineering Technion Israel Institute of Technology
More informationLecture 4 Noisy Channel Coding
Lecture 4 Noisy Channel Coding I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw October 9, 2015 1 / 56 I-Hsiang Wang IT Lecture 4 The Channel Coding Problem
More informationChannel Polarization and Blackwell Measures
Channel Polarization Blackwell Measures Maxim Raginsky Abstract The Blackwell measure of a binary-input channel (BIC is the distribution of the posterior probability of 0 under the uniform input distribution
More informationOn Improved Bounds for Probability Metrics and f- Divergences
IRWIN AND JOAN JACOBS CENTER FOR COMMUNICATION AND INFORMATION TECHNOLOGIES On Improved Bounds for Probability Metrics and f- Divergences Igal Sason CCIT Report #855 March 014 Electronics Computers Communications
More informationAn Improved Sphere-Packing Bound for Finite-Length Codes over Symmetric Memoryless Channels
POST-PRIT OF THE IEEE TRAS. O IFORMATIO THEORY, VOL. 54, O. 5, PP. 96 99, MAY 8 An Improved Sphere-Packing Bound for Finite-Length Codes over Symmetric Memoryless Channels Gil Wiechman Igal Sason Department
More informationOn the Block Error Probability of LP Decoding of LDPC Codes
On the Block Error Probability of LP Decoding of LDPC Codes Ralf Koetter CSL and Dept. of ECE University of Illinois at Urbana-Champaign Urbana, IL 680, USA koetter@uiuc.edu Pascal O. Vontobel Dept. of
More informationTime-invariant LDPC convolutional codes
Time-invariant LDPC convolutional codes Dimitris Achlioptas, Hamed Hassani, Wei Liu, and Rüdiger Urbanke Department of Computer Science, UC Santa Cruz, USA Email: achlioptas@csucscedu Department of Computer
More informationEECS 750. Hypothesis Testing with Communication Constraints
EECS 750 Hypothesis Testing with Communication Constraints Name: Dinesh Krithivasan Abstract In this report, we study a modification of the classical statistical problem of bivariate hypothesis testing.
More informationOn the Typicality of the Linear Code Among the LDPC Coset Code Ensemble
5 Conference on Information Sciences and Systems The Johns Hopkins University March 16 18 5 On the Typicality of the Linear Code Among the LDPC Coset Code Ensemble C.-C. Wang S.R. Kulkarni and H.V. Poor
More informationECEN 655: Advanced Channel Coding
ECEN 655: Advanced Channel Coding Course Introduction Henry D. Pfister Department of Electrical and Computer Engineering Texas A&M University ECEN 655: Advanced Channel Coding 1 / 19 Outline 1 History
More informationLecture 12. Block Diagram
Lecture 12 Goals Be able to encode using a linear block code Be able to decode a linear block code received over a binary symmetric channel or an additive white Gaussian channel XII-1 Block Diagram Data
More informationLecture 14 February 28
EE/Stats 376A: Information Theory Winter 07 Lecture 4 February 8 Lecturer: David Tse Scribe: Sagnik M, Vivek B 4 Outline Gaussian channel and capacity Information measures for continuous random variables
More informationTHIS paper provides a general technique for constructing
Protograph-Based Raptor-Like LDPC Codes for the Binary Erasure Channel Kasra Vakilinia Department of Electrical Engineering University of California, Los Angeles Los Angeles, California 90024 Email: vakiliniak@ucla.edu
More informationLOW-density parity-check (LDPC) codes were invented
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 54, NO 1, JANUARY 2008 51 Extremal Problems of Information Combining Yibo Jiang, Alexei Ashikhmin, Member, IEEE, Ralf Koetter, Senior Member, IEEE, and Andrew
More informationarxiv: v4 [cs.it] 8 Apr 2014
1 On Improved Bounds for Probability Metrics and f-divergences Igal Sason Abstract Derivation of tight bounds for probability metrics and f-divergences is of interest in information theory and statistics.
More informationGraph-based Codes for Quantize-Map-and-Forward Relaying
20 IEEE Information Theory Workshop Graph-based Codes for Quantize-Map-and-Forward Relaying Ayan Sengupta, Siddhartha Brahma, Ayfer Özgür, Christina Fragouli and Suhas Diggavi EPFL, Switzerland, UCLA,
More informationChannel Dispersion and Moderate Deviations Limits for Memoryless Channels
Channel Dispersion and Moderate Deviations Limits for Memoryless Channels Yury Polyanskiy and Sergio Verdú Abstract Recently, Altug and Wagner ] posed a question regarding the optimal behavior of the probability
More informationAn Introduction to Algorithmic Coding Theory
An Introduction to Algorithmic Coding Theory M. Amin Shokrollahi Bell Laboratories Part : Codes - A puzzle What do the following problems have in common? 2 Problem : Information Transmission MESSAGE G
More informationarxiv:cs/ v1 [cs.it] 16 Aug 2005
On Achievable Rates and Complexity of LDPC Codes for Parallel Channels with Application to Puncturing Igal Sason Gil Wiechman arxiv:cs/587v [cs.it] 6 Aug 5 Technion Israel Institute of Technology Haifa
More informationDispersion of the Gilbert-Elliott Channel
Dispersion of the Gilbert-Elliott Channel Yury Polyanskiy Email: ypolyans@princeton.edu H. Vincent Poor Email: poor@princeton.edu Sergio Verdú Email: verdu@princeton.edu Abstract Channel dispersion plays
More informationUnified Scaling of Polar Codes: Error Exponent, Scaling Exponent, Moderate Deviations, and Error Floors
Unified Scaling of Polar Codes: Error Exponent, Scaling Exponent, Moderate Deviations, and Error Floors Marco Mondelli, S. Hamed Hassani, and Rüdiger Urbanke Abstract Consider transmission of a polar code
More informationLow-Density Parity-Check Codes
Department of Computer Sciences Applied Algorithms Lab. July 24, 2011 Outline 1 Introduction 2 Algorithms for LDPC 3 Properties 4 Iterative Learning in Crowds 5 Algorithm 6 Results 7 Conclusion PART I
More informationVariable-Rate Universal Slepian-Wolf Coding with Feedback
Variable-Rate Universal Slepian-Wolf Coding with Feedback Shriram Sarvotham, Dror Baron, and Richard G. Baraniuk Dept. of Electrical and Computer Engineering Rice University, Houston, TX 77005 Abstract
More informationTight Bounds for Symmetric Divergence Measures and a New Inequality Relating f-divergences
Tight Bounds for Symmetric Divergence Measures and a New Inequality Relating f-divergences Igal Sason Department of Electrical Engineering Technion, Haifa 3000, Israel E-mail: sason@ee.technion.ac.il Abstract
More informationTight Bounds for Symmetric Divergence Measures and a Refined Bound for Lossless Source Coding
APPEARS IN THE IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY 015 1 Tight Bounds for Symmetric Divergence Measures and a Refined Bound for Lossless Source Coding Igal Sason Abstract Tight bounds for
More informationLecture 11: Polar codes construction
15-859: Information Theory and Applications in TCS CMU: Spring 2013 Lecturer: Venkatesan Guruswami Lecture 11: Polar codes construction February 26, 2013 Scribe: Dan Stahlke 1 Polar codes: recap of last
More informationGeneralized Writing on Dirty Paper
Generalized Writing on Dirty Paper Aaron S. Cohen acohen@mit.edu MIT, 36-689 77 Massachusetts Ave. Cambridge, MA 02139-4307 Amos Lapidoth lapidoth@isi.ee.ethz.ch ETF E107 ETH-Zentrum CH-8092 Zürich, Switzerland
More informationEE5139R: Problem Set 7 Assigned: 30/09/15, Due: 07/10/15
EE5139R: Problem Set 7 Assigned: 30/09/15, Due: 07/10/15 1. Cascade of Binary Symmetric Channels The conditional probability distribution py x for each of the BSCs may be expressed by the transition probability
More informationHypothesis Testing with Communication Constraints
Hypothesis Testing with Communication Constraints Dinesh Krithivasan EECS 750 April 17, 2006 Dinesh Krithivasan (EECS 750) Hyp. testing with comm. constraints April 17, 2006 1 / 21 Presentation Outline
More informationAnalysis of Sum-Product Decoding of Low-Density Parity-Check Codes Using a Gaussian Approximation
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 2, FEBRUARY 2001 657 Analysis of Sum-Product Decoding of Low-Density Parity-Check Codes Using a Gaussian Approximation Sae-Young Chung, Member, IEEE,
More informationArimoto-Rényi Conditional Entropy. and Bayesian M-ary Hypothesis Testing. Abstract
Arimoto-Rényi Conditional Entropy and Bayesian M-ary Hypothesis Testing Igal Sason Sergio Verdú Abstract This paper gives upper and lower bounds on the minimum error probability of Bayesian M-ary hypothesis
More informationIntroduction to Low-Density Parity Check Codes. Brian Kurkoski
Introduction to Low-Density Parity Check Codes Brian Kurkoski kurkoski@ice.uec.ac.jp Outline: Low Density Parity Check Codes Review block codes History Low Density Parity Check Codes Gallager s LDPC code
More informationBounds on Mutual Information for Simple Codes Using Information Combining
ACCEPTED FOR PUBLICATION IN ANNALS OF TELECOMM., SPECIAL ISSUE 3RD INT. SYMP. TURBO CODES, 003. FINAL VERSION, AUGUST 004. Bounds on Mutual Information for Simple Codes Using Information Combining Ingmar
More informationSecret Key and Private Key Constructions for Simple Multiterminal Source Models
Secret Key and Private Key Constructions for Simple Multiterminal Source Models arxiv:cs/05050v [csit] 3 Nov 005 Chunxuan Ye Department of Electrical and Computer Engineering and Institute for Systems
More informationUpper Bounds on the Capacity of Binary Intermittent Communication
Upper Bounds on the Capacity of Binary Intermittent Communication Mostafa Khoshnevisan and J. Nicholas Laneman Department of Electrical Engineering University of Notre Dame Notre Dame, Indiana 46556 Email:{mhoshne,
More informationCapacity of the Discrete Memoryless Energy Harvesting Channel with Side Information
204 IEEE International Symposium on Information Theory Capacity of the Discrete Memoryless Energy Harvesting Channel with Side Information Omur Ozel, Kaya Tutuncuoglu 2, Sennur Ulukus, and Aylin Yener
More informationSpatially Coupled LDPC Codes
Spatially Coupled LDPC Codes Kenta Kasai Tokyo Institute of Technology 30 Aug, 2013 We already have very good codes. Efficiently-decodable asymptotically capacity-approaching codes Irregular LDPC Codes
More informationAn Achievable Error Exponent for the Mismatched Multiple-Access Channel
An Achievable Error Exponent for the Mismatched Multiple-Access Channel Jonathan Scarlett University of Cambridge jms265@camacuk Albert Guillén i Fàbregas ICREA & Universitat Pompeu Fabra University of
More informationComputing the threshold shift for general channels
Computing the threshold shift for general channels Jeremie Ezri and Rüdiger Urbanke LTHC EPFL jeremie.ezri@epfl.ch, ruediger.urbanke@epfl.ch Andrea Montanari and Sewoong Oh Electrical Engineering Department
More informationLecture 6 I. CHANNEL CODING. X n (m) P Y X
6- Introduction to Information Theory Lecture 6 Lecturer: Haim Permuter Scribe: Yoav Eisenberg and Yakov Miron I. CHANNEL CODING We consider the following channel coding problem: m = {,2,..,2 nr} Encoder
More informationLecture 22: Final Review
Lecture 22: Final Review Nuts and bolts Fundamental questions and limits Tools Practical algorithms Future topics Dr Yao Xie, ECE587, Information Theory, Duke University Basics Dr Yao Xie, ECE587, Information
More informationQuantum Sphere-Packing Bounds and Moderate Deviation Analysis for Classical-Quantum Channels
Quantum Sphere-Packing Bounds and Moderate Deviation Analysis for Classical-Quantum Channels (, ) Joint work with Min-Hsiu Hsieh and Marco Tomamichel Hao-Chung Cheng University of Technology Sydney National
More informationEntropy and Ergodic Theory Lecture 15: A first look at concentration
Entropy and Ergodic Theory Lecture 15: A first look at concentration 1 Introduction to concentration Let X 1, X 2,... be i.i.d. R-valued RVs with common distribution µ, and suppose for simplicity that
More informationOn the Shamai-Laroia Approximation for the Information Rate of the ISI Channel
On the Shamai-Laroia Approximation for the Information Rate of the ISI Channel Yair Carmon and Shlomo Shamai (Shitz) Department of Electrical Engineering, Technion - Israel Institute of Technology 2014
More informationLECTURE 10. Last time: Lecture outline
LECTURE 10 Joint AEP Coding Theorem Last time: Error Exponents Lecture outline Strong Coding Theorem Reading: Gallager, Chapter 5. Review Joint AEP A ( ɛ n) (X) A ( ɛ n) (Y ) vs. A ( ɛ n) (X, Y ) 2 nh(x)
More informationSuperposition Encoding and Partial Decoding Is Optimal for a Class of Z-interference Channels
Superposition Encoding and Partial Decoding Is Optimal for a Class of Z-interference Channels Nan Liu and Andrea Goldsmith Department of Electrical Engineering Stanford University, Stanford CA 94305 Email:
More informationDistributed Source Coding Using LDPC Codes
Distributed Source Coding Using LDPC Codes Telecommunications Laboratory Alex Balatsoukas-Stimming Technical University of Crete May 29, 2010 Telecommunications Laboratory (TUC) Distributed Source Coding
More informationLecture 4 : Introduction to Low-density Parity-check Codes
Lecture 4 : Introduction to Low-density Parity-check Codes LDPC codes are a class of linear block codes with implementable decoders, which provide near-capacity performance. History: 1. LDPC codes were
More informationApproaching Blokh-Zyablov Error Exponent with Linear-Time Encodable/Decodable Codes
Approaching Blokh-Zyablov Error Exponent with Linear-Time Encodable/Decodable Codes 1 Zheng Wang, Student Member, IEEE, Jie Luo, Member, IEEE arxiv:0808.3756v1 [cs.it] 27 Aug 2008 Abstract We show that
More informationThe PPM Poisson Channel: Finite-Length Bounds and Code Design
August 21, 2014 The PPM Poisson Channel: Finite-Length Bounds and Code Design Flavio Zabini DEI - University of Bologna and Institute for Communications and Navigation German Aerospace Center (DLR) Balazs
More informationAn Efficient Maximum Likelihood Decoding of LDPC Codes Over the Binary Erasure Channel
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO. 11, NOVEMBER 24 1 An Efficient Maximum Likelihood Decoding of LDPC Codes Over the Binary Erasure Channel David Burshtein and Gadi Miller School of Electrical
More informationLow-complexity error correction in LDPC codes with constituent RS codes 1
Eleventh International Workshop on Algebraic and Combinatorial Coding Theory June 16-22, 2008, Pamporovo, Bulgaria pp. 348-353 Low-complexity error correction in LDPC codes with constituent RS codes 1
More informationSHARED INFORMATION. Prakash Narayan with. Imre Csiszár, Sirin Nitinawarat, Himanshu Tyagi, Shun Watanabe
SHARED INFORMATION Prakash Narayan with Imre Csiszár, Sirin Nitinawarat, Himanshu Tyagi, Shun Watanabe 2/40 Acknowledgement Praneeth Boda Himanshu Tyagi Shun Watanabe 3/40 Outline Two-terminal model: Mutual
More informationIterative Encoding of Low-Density Parity-Check Codes
Iterative Encoding of Low-Density Parity-Check Codes David Haley, Alex Grant and John Buetefuer Institute for Telecommunications Research University of South Australia Mawson Lakes Blvd Mawson Lakes SA
More informationBounds on the Maximum Likelihood Decoding Error Probability of Low Density Parity Check Codes
Bounds on the Maximum ikelihood Decoding Error Probability of ow Density Parity Check Codes Gadi Miller and David Burshtein Dept. of Electrical Engineering Systems Tel-Aviv University Tel-Aviv 69978, Israel
More informationELEC546 Review of Information Theory
ELEC546 Review of Information Theory Vincent Lau 1/1/004 1 Review of Information Theory Entropy: Measure of uncertainty of a random variable X. The entropy of X, H(X), is given by: If X is a discrete random
More informationPractical Polar Code Construction Using Generalised Generator Matrices
Practical Polar Code Construction Using Generalised Generator Matrices Berksan Serbetci and Ali E. Pusane Department of Electrical and Electronics Engineering Bogazici University Istanbul, Turkey E-mail:
More informationJoint Iterative Decoding of LDPC Codes and Channels with Memory
Joint Iterative Decoding of LDPC Codes and Channels with Memory Henry D. Pfister and Paul H. Siegel University of California, San Diego 3 rd Int l Symposium on Turbo Codes September 1, 2003 Outline Channels
More informationExact Probability of Erasure and a Decoding Algorithm for Convolutional Codes on the Binary Erasure Channel
Exact Probability of Erasure and a Decoding Algorithm for Convolutional Codes on the Binary Erasure Channel Brian M. Kurkoski, Paul H. Siegel, and Jack K. Wolf Department of Electrical and Computer Engineering
More informationWhy We Can Not Surpass Capacity: The Matching Condition
Why We Can Not Surpass Capacity: The Matching Condition Cyril Méasson EPFL CH-1015 Lausanne cyril.measson@epfl.ch, Andrea Montanari LPT, ENS F-75231 Paris montanar@lpt.ens.fr Rüdiger Urbanke EPFL CH-1015
More informationAn Introduction to Low-Density Parity-Check Codes
An Introduction to Low-Density Parity-Check Codes Paul H. Siegel Electrical and Computer Engineering University of California, San Diego 5/ 3/ 7 Copyright 27 by Paul H. Siegel Outline Shannon s Channel
More informationCommunications Theory and Engineering
Communications Theory and Engineering Master's Degree in Electronic Engineering Sapienza University of Rome A.A. 2018-2019 AEP Asymptotic Equipartition Property AEP In information theory, the analog of
More informationTHE seminal paper of Gallager [1, p. 48] suggested to evaluate
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 11, NOVEMBER 2004 2657 Extrinsic Information Transfer Functions: Model and Erasure Channel Properties Alexei Ashikhmin, Member, IEEE, Gerhard Kramer,
More informationOn Bit Error Rate Performance of Polar Codes in Finite Regime
On Bit Error Rate Performance of Polar Codes in Finite Regime A. Eslami and H. Pishro-Nik Abstract Polar codes have been recently proposed as the first low complexity class of codes that can provably achieve
More informationAccumulate-Repeat-Accumulate Codes: Capacity-Achieving Ensembles of Systematic Codes for the Erasure Channel with Bounded Complexity
Accumulate-Repeat-Accumulate Codes: Capacity-Achieving Ensembles of Systematic Codes for the Erasure Channel with Bounded Complexity Henry D. Pfister, Member, Igal Sason, Member Abstract The paper introduces
More information5. Density evolution. Density evolution 5-1
5. Density evolution Density evolution 5-1 Probabilistic analysis of message passing algorithms variable nodes factor nodes x1 a x i x2 a(x i ; x j ; x k ) x3 b x4 consider factor graph model G = (V ;
More informationSecond-Order Asymptotics in Information Theory
Second-Order Asymptotics in Information Theory Vincent Y. F. Tan (vtan@nus.edu.sg) Dept. of ECE and Dept. of Mathematics National University of Singapore (NUS) National Taiwan University November 2015
More informationEntropy, Inference, and Channel Coding
Entropy, Inference, and Channel Coding Sean Meyn Department of Electrical and Computer Engineering University of Illinois and the Coordinated Science Laboratory NSF support: ECS 02-17836, ITR 00-85929
More informationA t super-channel. trellis code and the channel. inner X t. Y t. S t-1. S t. S t+1. stages into. group two. one stage P 12 / 0,-2 P 21 / 0,2
Capacity Approaching Signal Constellations for Channels with Memory Λ Aleksandar Kav»cić, Xiao Ma, Michael Mitzenmacher, and Nedeljko Varnica Division of Engineering and Applied Sciences Harvard University
More informationOn Third-Order Asymptotics for DMCs
On Third-Order Asymptotics for DMCs Vincent Y. F. Tan Institute for Infocomm Research (I R) National University of Singapore (NUS) January 0, 013 Vincent Tan (I R and NUS) Third-Order Asymptotics for DMCs
More informationInteractions of Information Theory and Estimation in Single- and Multi-user Communications
Interactions of Information Theory and Estimation in Single- and Multi-user Communications Dongning Guo Department of Electrical Engineering Princeton University March 8, 2004 p 1 Dongning Guo Communications
More informationInformation Theory and Hypothesis Testing
Summer School on Game Theory and Telecommunications Campione, 7-12 September, 2014 Information Theory and Hypothesis Testing Mauro Barni University of Siena September 8 Review of some basic results linking
More informationOn the Design of Raptor Codes for Binary-Input Gaussian Channels
1 On the Design of Raptor Codes for Binary-Input Gaussian Channels Zhong Cheng, Jeff Castura, and Yongyi Mao, Member, IEEE Abstract This paper studies the problem of Raptor-code design for binary-input
More informationArimoto Channel Coding Converse and Rényi Divergence
Arimoto Channel Coding Converse and Rényi Divergence Yury Polyanskiy and Sergio Verdú Abstract Arimoto proved a non-asymptotic upper bound on the probability of successful decoding achievable by any code
More informationBelief propagation decoding of quantum channels by passing quantum messages
Belief propagation decoding of quantum channels by passing quantum messages arxiv:67.4833 QIP 27 Joseph M. Renes lempelziv@flickr To do research in quantum information theory, pick a favorite text on classical
More information