Two Applications of the Gaussian Poincaré Inequality in the Shannon Theory

Transcription

1 Two Applications of the Gaussian Poincaré Inequality in the Shannon Theory Vincent Y. F. Tan (Joint work with Silas L. Fong) National University of Singapore (NUS) 2016 International Zurich Seminar on Communications Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

2 Gaussian Poincaré Inequality Theorem For Z n iid N (0, 1) and any differentiable mapping f such that E[(f (Z n )) 2 ] <, and E[ f (Z n ) 2 ] < we have var[f (Z n )] E[ f (Z n ) 2 ]. Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

3 Gaussian Poincaré Inequality Theorem For Z n iid N (0, 1) and any differentiable mapping f such that E[(f (Z n )) 2 ] <, and E[ f (Z n ) 2 ] < we have var[f (Z n )] E[ f (Z n ) 2 ]. Controlling the variance of f (Z n ) that is a function of i.i.d. random variables in terms of the gradient of f (Z n ) Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

4 Gaussian Poincaré Inequality Theorem For Z n iid N (0, 1) and any differentiable mapping f such that E[(f (Z n )) 2 ] <, and E[ f (Z n ) 2 ] < we have var[f (Z n )] E[ f (Z n ) 2 ]. Controlling the variance of f (Z n ) that is a function of i.i.d. random variables in terms of the gradient of f (Z n ) Using the Gaussian Poincaré inequality for appropriate f, var[f (Z n )] = O(n). Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

5 Gaussian Poincaré Inequality in Shannon Theory Polyanskiy and Verdú (2014) bounded the KL divergence between the empirical output distribution of AWGN channel codes P Y n and the n-fold product of the CAOD P Y, i.e., D(P Y n (P Y) n ) Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

6 Gaussian Poincaré Inequality in Shannon Theory Polyanskiy and Verdú (2014) bounded the KL divergence between the empirical output distribution of AWGN channel codes P Y n and the n-fold product of the CAOD P Y, i.e., D(P Y n (P Y) n ) Often we need to bound the variance of certain log-likelihood ratios (dispersion) Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

7 Gaussian Poincaré Inequality in Shannon Theory Polyanskiy and Verdú (2014) bounded the KL divergence between the empirical output distribution of AWGN channel codes P Y n and the n-fold product of the CAOD P Y, i.e., D(P Y n (P Y) n ) Often we need to bound the variance of certain log-likelihood ratios (dispersion) Demonstrate its utility by establishing Strong converse for the Gaussian broadcast channels Properties of the empirical output distribution of delay-limited codes for quasi-static fading channels Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

8 Gaussian Broadcast Channel Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

9 Gaussian Broadcast Channel Assume g 1 = g 2 = 1 and σ 2 2 > σ2 1 Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

10 Gaussian Broadcast Channel Assume g 1 = g 2 = 1 and σ 2 2 > σ2 1 Input X n must satisfy X n 2 2 = n Xi 2 np i=1 Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

11 Capacity Region An (n, M 1n, M 2n, ε n )-code consists of an encoder f : {1,..., M 1n } {1,..., M 2n } R n such that the power constraint is satisfied; two decoders ϕ j : R n {1,..., M jn } for j = 1, 2; such that the average error probability P (n) e := Pr(Ŵ 1 W 1 or Ŵ 2 W 2 ) ε n. Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

12 Capacity Region An (n, M 1n, M 2n, ε n )-code consists of an encoder f : {1,..., M 1n } {1,..., M 2n } R n such that the power constraint is satisfied; two decoders ϕ j : R n {1,..., M jn } for j = 1, 2; such that the average error probability P (n) e := Pr(Ŵ 1 W 1 or Ŵ 2 W 2 ) ε n. (R 1, R 2 ) is achievable a sequence of (n, M 1n, M 2n, ε n )-codes s.t. lim inf n 1 n log M jn R j, j = 1, 2, and lim n = 0. n Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

13 Capacity Region An (n, M 1n, M 2n, ε n )-code consists of an encoder f : {1,..., M 1n } {1,..., M 2n } R n such that the power constraint is satisfied; two decoders ϕ j : R n {1,..., M jn } for j = 1, 2; such that the average error probability P (n) e := Pr(Ŵ 1 W 1 or Ŵ 2 W 2 ) ε n. (R 1, R 2 ) is achievable a sequence of (n, M 1n, M 2n, ε n )-codes s.t. lim inf n 1 n log M jn R j, j = 1, 2, and lim n = 0. n Capacity region C is the set of all achievable rate pairs Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

14 Capacity Region Cover (1972) and Bergmans (1974) showed that C = R BC = R(α) α [0,1] where { ( αp ) ( (1 α)p ) } R(α) = (R 1, R 2 ) : R 1 C σ1 2, R 2 C αp + σ2 2 and C(x) = 1 log(1 + x). 2 Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

15 Capacity Region Cover (1972) and Bergmans (1974) showed that C = R BC = R(α) α [0,1] where { ( αp ) ( (1 α)p ) } R(α) = (R 1, R 2 ) : R 1 C σ1 2, R 2 C αp + σ2 2 and C(x) = 1 log(1 + x). 2 Direct part: Random coding + Superposition coding Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

16 Capacity Region Cover (1972) and Bergmans (1974) showed that C = R BC = R(α) α [0,1] where { ( αp ) ( (1 α)p ) } R(α) = (R 1, R 2 ) : R 1 C σ1 2, R 2 C αp + σ2 2 and C(x) = 1 log(1 + x). 2 Direct part: Random coding + Superposition coding Converse part: Fano s inequality + Entropy power inequality Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

17 Capacity Region Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

18 Capacity Region P (n) e 0 Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

19 Capacity Region P (n) e 0 P (n) e 0 Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

20 Capacity Region P (n) e 0 P (n) e 1? P (n) e 0 Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

21 Strong converse vs weak converse Can we claim that if (R 1, R 2 ) / C, then P (n) e 1? Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

22 Strong converse vs weak converse Can we claim that if (R 1, R 2 ) / C, then P (n) e 1? Indeed! Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

23 Strong converse vs weak converse Can we claim that if (R 1, R 2 ) / C, then P (n) e 1? Indeed! Sharp phase transition between what s possible and what s not Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

24 Strong converse vs weak converse Can we claim that if (R 1, R 2 ) / C, then P (n) e 1? Indeed! Sharp phase transition between what s possible and what s not The strong converse has been established for only degraded discrete memoryless BC Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

25 Strong converse vs weak converse Can we claim that if (R 1, R 2 ) / C, then P (n) e 1? Indeed! Sharp phase transition between what s possible and what s not The strong converse has been established for only degraded discrete memoryless BC Ahlswede, Gács and Körner (1976) used the blowing-up lemma BUL doesn t work for continuous alphabets [but see Wu and Özgür (2015)] Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

26 Strong converse vs weak converse Can we claim that if (R 1, R 2 ) / C, then P (n) e 1? Indeed! Sharp phase transition between what s possible and what s not The strong converse has been established for only degraded discrete memoryless BC Ahlswede, Gács and Körner (1976) used the blowing-up lemma BUL doesn t work for continuous alphabets [but see Wu and Özgür (2015)] Oohama (2015) uses properties of the Rényi divergence Good bounds between the Rényi divergence D α (P Q) and the relative entropy D(P Q) exist for finite alphabets Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

27 ε-capacity Region (R 1, R 2 ) is ε-achievable a sequence of (n, M 1n, M 2n, ε n )-codes s.t. lim inf n 1 n log M jn R j, j = 1, 2, and lim sup ε n ε. n Capacity region C ε is the set of all achievable rate pairs Strong converse holds iff C ε does not depend on ε. We already know that R BC = C 0 C ε Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

28 Strong converse Theorem The Gaussian BC satisfies the strong converse property: C ε = R BC, ε [0, 1) Key ideas in proof: Derive an appropriate information spectrum converse bound Use the Gaussian Poincaré inequality to bound relevant variances Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

29 Weak Converse for GBC [Bergmans (1974)] Step 1: Invoke Fano s inequality to assert that for any sequence of codes with vanishing error probability ε n 0, R j 1 n I(W j; Y n j ) + o(1), j {1, 2}. Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

30 Weak Converse for GBC [Bergmans (1974)] Step 1: Invoke Fano s inequality to assert that for any sequence of codes with vanishing error probability ε n 0, R j 1 n I(W j; Y n j ) + o(1), j {1, 2}. Step 2: Single-letterize and entropy power inequality ( ) I(W 1 ; Y1 n ) ni(x; Y 1 U) EPI αp nc ( ) I(W 1 ; Y1 n ) + I(W 2; Y2 n ) ni(u; Y 2) EPI (1 α)p nc αp + σ2 2 σ 2 1 Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

31 Strong Converse for DM-BC [Ahlswede et al. (1976)] Step 1: Invoke the blowing-up lemma to assert that for any sequence of codes with non-vanishing error probability ε [0, 1), R j 1 n I(W j; Y n j ) + o(1), j {1, 2}. Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

32 Strong Converse for DM-BC [Ahlswede et al. (1976)] Step 1: Invoke the blowing-up lemma to assert that for any sequence of codes with non-vanishing error probability ε [0, 1), R j 1 n I(W j; Y n j ) + o(1), j {1, 2}. Step 2: Single-letterize I(W 1 ; Y n 1 ) ni(x; Y 1 U), I(W 1 ; Y n 1 ) + I(W 2; Y n 2 ) ni(u; Y 2) where U i := (W 2, Y i 1 1 ). One also uses the degradedness condition here: I(W 2, Y i 1 2, Y i 1 1 ; Y 2i ) = I(U i ; Y 2i ). Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

33 Our Strong Converse Proof for Gaussian BC Convert code defined based on avg error prob ε to one based on max error prob ε =: ε w/o loss in rate [Telatar] Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

34 Our Strong Converse Proof for Gaussian BC Convert code defined based on avg error prob ε to one based on max error prob ε =: ε w/o loss in rate [Telatar] Establish information spectrum bound. For every (w 1, w 2 ), every code with max error prob ε satisfies ( ε Pr log P(Yn 1 w ) 1) P(Y1 n) nr 1 γ 1 (w 1, w 2 ) n 2 e γ 1(w 1,w 2 ) { } 1 2 n(r 1+R 2 ) P(y n 1 )P(w 2 w 1, y n 1 ) dyn 1 > n2 D 1 (w 1 ) Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

35 Our Strong Converse Proof for Gaussian BC Convert code defined based on avg error prob ε to one based on max error prob ε =: ε w/o loss in rate [Telatar] Establish information spectrum bound. For every (w 1, w 2 ), every code with max error prob ε satisfies ( ε Pr log P(Yn 1 w ) 1) P(Y1 n) nr 1 γ 1 (w 1, w 2 ) n 2 e γ 1(w 1,w 2 ) { } 1 2 n(r 1+R 2 ) P(y n 1 )P(w 2 w 1, y n 1 ) dyn 1 > n2 D 1 (w 1 ) Indicator term is often negligible (by Markov s inequality) Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

36 Our Strong Converse Proof for Gaussian BC Convert code defined based on avg error prob ε to one based on max error prob ε =: ε w/o loss in rate [Telatar] Establish information spectrum bound. For every (w 1, w 2 ), every code with max error prob ε satisfies ( ε Pr log P(Yn 1 w ) 1) P(Y1 n) nr 1 γ 1 (w 1, w 2 ) n 2 e γ 1(w 1,w 2 ) { } 1 2 n(r 1+R 2 ) P(y n 1 )P(w 2 w 1, y n 1 ) dyn 1 > n2 D 1 (w 1 ) Indicator term is often negligible (by Markov s inequality) Establish a bound on the coding rate [ nr 1 E log P(Yn 1 w ] 1) P(Y1 n) ε var [ log P(Yn 1 w ] 1) P(Y1 n) + 3 log n Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

37 Our Strong Converse Proof for Gaussian BC Use the Gaussian Poincaré inequality with careful identification of f and peak power constraint X n 2 np to assert that [ var log P(Yn 1 w ] 1) P(Y1 n) = O(n). Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

38 Our Strong Converse Proof for Gaussian BC Use the Gaussian Poincaré inequality with careful identification of f and peak power constraint X n 2 np to assert that [ var log P(Yn 1 w ] 1) P(Y1 n) = O(n). Thus we conclude that nr 1 I(W 1 ; Y n 1 ) + O( n), ε [0, 1). Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

39 Our Strong Converse Proof for Gaussian BC Use the Gaussian Poincaré inequality with careful identification of f and peak power constraint X n 2 np to assert that [ var log P(Yn 1 w ] 1) P(Y1 n) = O(n). Thus we conclude that nr 1 I(W 1 ; Y n 1 ) + O( n), ε [0, 1). Backoff term is of the order O(1/ n), i.e., where λ log M 1n + (1 λ) log M 2n = nc λ + O( n) { ( αp ) ( (1 α)p ) } C λ := max λc α [0,1] σ1 2 + (1 λ)c αp + σ2 2 but nailing down the constant (dispersion) seems challenging. Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

40 Another Application: Quasi-Static Fading Channels Consider the channel model Y i = HX i + Z i, i = 1,..., n where Z i are independent standard normal random variables and E[1/H] (0, ) Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

41 Another Application: Quasi-Static Fading Channels Consider the channel model Y i = HX i + Z i, i = 1,..., n where Z i are independent standard normal random variables and E[1/H] (0, ) If fading state is h and message is w, codeword is f h (w) R n. Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

42 Another Application: Quasi-Static Fading Channels Consider the channel model Y i = HX i + Z i, i = 1,..., n where Z i are independent standard normal random variables and E[1/H] (0, ) If fading state is h and message is w, codeword is f h (w) R n. Long-term power constraint 1 M n M n w=1 R + P H (h) f h (w) 2 dh np Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

43 Another Application: Quasi-Static Fading Channels Consider the channel model Y i = HX i + Z i, i = 1,..., n where Z i are independent standard normal random variables and E[1/H] (0, ) If fading state is h and message is w, codeword is f h (w) R n. Long-term power constraint 1 M n M n w=1 R + P H (h) f h (w) 2 dh np Delay-limited capacity [Hanly and Tse (1998)], i.e., the maximum transmission rate under the constraint that the maximal error probability over all H > 0 vanishes Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

44 Vanishing Normalized Relative Entropy The delay-limited capacity [Hanly and Tse (1998)] is C(P DL ), where P DL := P E[1/H] For any sequence of capacity-achieving codes with vanishing maximum error probability 1 lim n n D( P Y n (P Y) n) 0 where P Y = N (0, P DL ). Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

45 Vanishing Normalized Relative Entropy The delay-limited capacity [Hanly and Tse (1998)] is C(P DL ), where P DL := P E[1/H] For any sequence of capacity-achieving codes with vanishing maximum error probability 1 lim n n D( P Y n (P Y) n) 0 where P Y = N (0, P DL ). Every good code is s.t. the induced output distribution looks like the n-fold CAOD. Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

46 Vanishing Normalized Relative Entropy The delay-limited capacity [Hanly and Tse (1998)] is C(P DL ), where P DL := P E[1/H] For any sequence of capacity-achieving codes with vanishing maximum error probability 1 lim n n D( P Y n (P Y) n) 0 where P Y = N (0, P DL ). Every good code is s.t. the induced output distribution looks like the n-fold CAOD. Extend to the case where the error probability is non-vanishing Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

47 Vanishing Normalized Relative Entropy The delay-limited capacity [Hanly and Tse (1998)] is C(P DL ), where P DL := P E[1/H] For any sequence of capacity-achieving codes with vanishing maximum error probability 1 lim n n D( P Y n (P Y) n) 0 where P Y = N (0, P DL ). Every good code is s.t. the induced output distribution looks like the n-fold CAOD. Extend to the case where the error probability is non-vanishing Control a variance term [ var log P Y n X n,h(y n X n ], h) P Y n H(Y n = O(n). h) Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

48 Concluding Remarks Gaussian Poincaré inequality is useful for Shannon-theoretic problems with uncountable alphabets var[f (Z n )] E[ f (Z n ) 2 ]. Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

49 Concluding Remarks Gaussian Poincaré inequality is useful for Shannon-theoretic problems with uncountable alphabets var[f (Z n )] E[ f (Z n ) 2 ]. Allows us to establish strong converses and properties of good codes by controlling variance of log-likelihood ratios Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17

50 Concluding Remarks Gaussian Poincaré inequality is useful for Shannon-theoretic problems with uncountable alphabets var[f (Z n )] E[ f (Z n ) 2 ]. Allows us to establish strong converses and properties of good codes by controlling variance of log-likelihood ratios For more details, please see Arxiv: (Strong converse for Gaussian broadcast) Arxiv: (Empirical output distribution of good codes for fading channels) Vincent Tan (NUS) Applications of the Poincaré Inequality IZS / 17