Post on 09-Jul-2020
transcript
How to Solve Gaussian Interference
Channel
WPI, HKU, 2019
Fan Cheng
Shanghai Jiao Tong University
chengfan@sjtu.edu.cn
β π + π‘π
π2
2ππ₯2π π₯, π‘ =
π
ππ‘π(π₯, π‘)
β π = ββ« π₯logπ₯ dπ₯π = π + π‘ππ βΌ π©(0,1)
β‘ A new mathematical theory on Gaussian distributionβ‘ Its application on Gaussian interference channelβ‘ History, progress, and future
β‘ History of βSuper-Hβ Theorem
β‘ Boltzmann equation, heat equation
β‘ Shannon Entropy Power Inequality
β‘ Complete Monotonicity Conjecture
β‘ How to Solve Gaussian Interference Channel
Outline
Fire and Civilization
DrillSteam engine
James WattsMyth: west and east
Independence of US
The Wealth of Nations
1776
Study of Heat
Heat transferβ‘ The history begins with the work of Joseph
Fourier around 1807β‘ In a remarkable memoir, Fourier invented
both Heat equation and the method of Fourier analysis for its solution
π
ππ‘π π₯, π‘ =
1
2
π2
ππ₯2π(π₯, π‘)
Information Age
ππ‘ βΌ π©(0, π‘)Gaussian Channel:
X and Z are mutually independent. The p.d.f of X is g(x)
ππ‘ is the convolution of X and ππ‘. ππ‘ β π + ππ‘
The probability density function (p.d.f.) of ππ‘
π(π¦; π‘) = β« π(π₯)1
2ππ‘π(π¦βπ₯)2
2π‘
π
ππ‘π(π¦; π‘) =
1
2
π2
ππ¦2π(π¦; π‘)
The p.d.f. of Y is the solution to the heat equation, and vice versa.
Gaussian channel and heat equation are identical in mathematics.
A mathematical theory of communication,
Bell System Technical Journal.
Ludwig Boltzmann
Boltzmann formula:
Boltzmann equation:
H-theorem:
Ludwig Eduard Boltzmann
1844-1906
Vienna, Austrian Empire
π = βππ΅lnππ = βππβ
πππlnππ
ππ
ππ‘= (
ππ
ππ‘)force + (
ππ
ππ‘)diff + (
ππ
ππ‘)coll
π»(π(π‘))is nonβdecreasing
Gibbs formula:
β‘ McKeanβs Problem on Boltzmann equation (1966): β‘ π»(π(π‘)) is CM in π‘, when
π π‘ satisfies Boltzmann equationβ‘ False, disproved by E. Lieb in
1970sβ‘ the particular Bobylev-Krook-Wu
explicit solutions, this βtheoremβ holds true for π β€ 101 and breaks downs afterwards
βSuper H-theoremβ for Boltzmann Equation
H. P. McKean, NYU.
National Academy of Sciences
A function is completely monotone (CM) iff all the signs of its derivatives
are alternating in +/-: +, -, +, -,β¦β¦ (e.g., 1/π‘, πβπ‘ )
β‘ Heat equation: Is π»(π(π‘)) CM in π‘, if π(π‘) satisfies heat equation
β‘ Equivalently, is π»(π + π‘π) CM in t? β‘ The signs of the first two order derivatives were obtainedβ‘ Failed to obtain the 3rd and 4th. (It is easy to compute the
derivatives, it is hard to obtain their signs)
βSuper H-theoremβ for Heat Equation
βThis suggests thatβ¦β¦, etc., but I could not prove itβ
-- H. P. McKean
C. Villani, 2010 Fields Medalist
Claude E. Shannon and EPI
β‘ Entropy power inequality (Shannon 1948): For any two independent continuous random variables X and Y,
Equality holds iff X and Y are Gaussianβ‘ Motivation: Gaussian noise is the worst noiseβ‘ Impact: A new characterization of Gaussian distribution in
information theoryβ‘ Comments: most profound! (Kolmogorov)
π2β(πΏ+π) β₯ π2β(πΏ) + π2β(π)
Central limit theoremCapacity region of Gaussian broadcast channelCapacity region of Gaussian Multiple-Input Multiple-Output broadcast channelUncertainty principle
All of them can be proved by Entropy Power Inequality (EPI)
β‘ Shannon himself didnβt give a proof but an explanation, which turned out to be wrong
β‘ The first proof is given by A. J. Stam (1959), N. M. Blachman (1966)
β‘ Research on EPIGeneralization, new proof, new connection. E.g., Gaussian interference channel is
open, some stronger βEPIββ should exist.
β‘ Stanford Information Theory School: Thomas Cover and his students: A. El Gamel, M. H. Costa, A. Dembo, A. Barron (1980-1990)
β‘ After 2000, Princeton && UC Berkeley
Entropy Power Inequality
Heart of Shannon theory
Ramification of EPI
Shannon EPI
Gaussian perturbation: β(π + π‘π)
Fisher Information: πΌ π + π‘π =π
ππ‘β(π + π‘π)/2
Fisher Information is decreasing in π‘
π2β(π+ π‘π) is concave in π‘Fisher information inequality (FII):
1
πΌ(π+π)β₯
1
πΌ(π)+
1
πΌ(π)
Tight Youngβs inequality
π + π π β₯ π π π π π
Status Quo: FII can imply EPI and all its generalizations.
Many network information problems remain open even
the noise is Gaussian.
--Only EPI is not sufficient
Where our journey begins Shannon Entropy power inequality
Fisher information inequality
β(π + π‘π)
β π π‘ is CM
When π(π‘) satisfied Boltzmann equation, disproved
When π(π‘) satisfied heat equation, unknown
We even donβt know what CM is!
Mathematician ignored it
Raymond introduced this paper to me in 2008
I made some progress with Chandra Nair in 2011 (MGL)
Complete monotonicity (CM) was discovered in 2012
The third derivative in 2013 (Key breakthrough)
The fourth order in 2014
Recently, CM GIC
Motivation
Motivation: to find some inequalities to obtain a better rate region; e.g., the
convexity of π(πΏ + πβππ), the concavity of π° πΏ+ ππ
π, etc.
βAny progress?β
βNopeβ¦β
It is widely believed that there should be no
new EPI except Shannon EPI and FII.
Observation: π°(πΏ + ππ) is convex in π
πΌ π + π‘π =π
2ππ‘β π + π‘π β₯ 0 (de Bruijn, 1958)
πΌ(1) =π
ππ‘πΌ π + π‘π β€ 0 (McKean1966, Costa 1985)
Could the third one be determined?
Discovery
Observation: π°(πΏ + ππ) is convex in π
β π + π‘π =1
2ln 2πππ‘, πΌ π + π‘π =
1
π‘. πΌ is CM: +, -, +, -β¦
If the observation is true, the first three derivatives are: +, -, +
Q: Is the 4th order derivative -? Because π is Gaussian! If so, thenβ¦
The signs of derivatives of β(π + π‘π) are independent of π. Invariant!
Exactly the same problem in McKeanβs 1966 paper
To convince people, must prove its convexity
My own opinion:
β’ A new fundamental result on Gaussian distribution
β’ Invariant is very important in mathematics
β’ In mathematics, the more beautiful, the more powerful
β’ Very hard to make any progress
Challenge
Let π βΌ π(π₯)
β ππ‘ = ββ« π(π¦, π‘) ln π(π¦, π‘) ππ¦: no closed-form expression
except for some special π π₯ . π(π¦, π‘) satisfies heat equation.
πΌ ππ‘ = β«π12
πππ¦
πΌ 1 ππ‘ = ββ«π2
πβ
π12
π2
2
ππ¦
So what is πΌ(2)? (Heat equation, integration by parts)
Challenge (contβd)
It is trivial to calculate derivatives.
It is not generally obvious to prove their signs.
π°
Breakthrough
Integration by parts: β« π’ππ£ = π’π£ β β« π£ππ’
First breakthrough since
McKean 1966
GCMCGaussian complete monotonicity conjecture (GCMC):
π°(πΏ + ππ) is CM in π
A general form: number partition. Hard to determine the coefficients.
Conjecture 2: π₯π¨π π°(πΏ + ππ) is convex in π
Hard to find π½π,π !
Moreover
C. Villani showed the work of H. P. McKean to us.
G. Toscani cited our work within two weeks:
the consequences of the evolution of the entropy and of its subsequent
derivatives along the solution to the heat equation have important consequences.
Indeed the argument of McKean about the signs of the first two derivatives are
equivalent to the proof of the logarithmic Sobolev inequality.
Gaussian optimality for derivatives of differential entropy using linear matrix inequalities
X. Zhang, V. Anantharam, Y. Geng - Entropy, 2018 - mdpi.com
β’ A new method to prove signs by LMI
β’ Verified the first four derivatives
β’ For the fifth order derivative, current methods cannot find a solution
Complete monotone function
Herbert R. Stahl, 2013
π π‘ = ΰΆ±0
β
πβπ‘π₯ ππ(π₯)
A new expression for entropy involved special
functions in mathematical physics
How to construct π(π₯)?
π
Complete monotone function
Theorem: A function π(π‘) is CM in π‘, then log π(π‘) is also convex in π‘ πΌ ππ‘ is CM in π‘, then log πΌ(ππ‘) is convex in π‘ (Conjecture 1 implies
Conjecture 2)
A function f(t) is CM, a Schur-convex function can be obtained by f(t) Schur-convex β Majority theory
Remarks: The current tools in information
theory donβt work. More sophisticated tools
should be built to attack this problem.
A new mathematical foundation of
information theory
1946
True Vs. False
If GCMC is true A fundamental breakthrough in mathematical physics, information
theory and any disciplines related to Gaussian distribution A new expression for Fisher information Derivatives are an invariant
Though β(π + π‘π) looks very messy, certain regularity exists Application: Gaussian interference channel?
If GCMC is false No Failure, as heat equation is a physical phenomenon A Gauss constant (e.g. 2019), where Gaussian distribution fails. Painful!
Complete Monotonicity:
How to Solve Gaussian Interference Channel
Two fundamental channel coding problem: BC and GIC
β ππ1 + ππ2 + π1 , β ππ1 + ππ2 + π2 exceed the
capability of EPI
Han-Kobayashi inner bound
Many researchers have contributed to this model
Foundation of wireless communication
The Thick Shell over β(π + π‘π)
β(π + π‘π) is hard to estimate:
The p.d.f of π + π‘π is messy
π π₯ log π(π₯) β« π π₯ logπ(π₯)No generally useful lower or upper bounds
--The thick shell over π + π‘π
Analysis: alternating is the worst
If the CM property of β(π + π‘π) is not true Take 5 for example: if CM breaks down after n=5 If we just take the 5th derivative, there may be nothing special.
(So GIC wonβt be so hard) CM affected the rate region of GIC
Prof. Siu, Yum-Tong: βAlternating is the worst thing in analysis as the integral is hard to converge, though CM is very beautifulβ It is not strange that Gaussian distribution is the worst in
information theory
Common viewpoint: information theory is about information inequality: EPI, MGL, etc.
CM is a class of inequalities. We should regard it as a whole in application. We should pivot our viewpoint from inequalities.
Information Decomposition
The lesson learned from complete monotonicity
πΌ π + π‘π = ΰΆ±0
β
πβπ‘π₯ππ(π₯)
Two independent components: πβπ‘π₯ stands for complete monotonicity
ππ(π₯) serves as the identity of πΌ π + π‘π Information decomposition:
Fisher Information = Complete Monotonicity + Borel Measure
CM is the thick shell. It can be used to estimate in majority theory Very useful in analysis and geometry
ππ(π₯) involves only π₯, and π‘ is removed The thick shell is removed from Fisher information ππ(π₯) is relatively easier to study than Fisher information WE know very little about ππ(π₯)
Only CM is useless for (network) information theory The current constraints on ππ(π₯) are too loose Only the βspecial oneβ is useful, otherwise every CM function should
have the same meaning in information theory
CM && GIC
A fundamental problem should have a nice and clean solution.
To understand complete monotonicity is not an easy job (10 years).
Top players are ready, but the football is missingβ¦
Thanks!
Guangyue
Raymond, Chandra, Venkat, Vincent...