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Random Matrix Theory Lecture 3 Free Probability Theory
Symeon Chatzinotas March 4, 2013 Luxembourg
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Outline
1. Free Probability Theory 1. Definitions
2. Asymptotically free matrices
3. R-transform
4. Additive Convolution
5. Sigma-transform
6. Multiplicative Convolution
2. Examples 1. Spectrum Sensing
2. Relay Channel
3. Cochannel interference
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Introduction
• Free probability theory
• A form of independence for non commutative algebras
• Applications in random Hermitian matrices
• Expressions that include sums or products of asymptotically free matrices
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Non-commutative Spaces
• Non commutative probability space (A,φ)
– A non-commutative unital algebra A
– A linear function φ:AC with φ(1)=1
– Moments
• Probability space (AN,τN)
– Random Hermitian Matrices AN
– Real random eigenvalues
– Functional τN
– τN(I)=1
– Moments τN(Xk)
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Asymptotic Freeness
• A family of matrices {XN,1,…, XN,n} is asymptotically free in (AN,τN) if:
– XN,n has a non random limit distribution
– For every family of polynomials where
It applies
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Asymptotically Free Matrices
• Any random matrix and the identity matrix
• Independent Wigner matrices
• Independent Gaussian matrices
• Independent Haar matrices
• Independent Unitarily Invariant (Wishart) matrices
• Standard Winger and deterministic diagonal
• Standard Gaussian and deterministic (diagonal)
• Haar matrices and deterministic matrix
• Unitarily invariant and deterministic matrix
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R Transform: Definition
• z belongs to the complex plane
• Reminder: Stieltjes definition
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R transform: Basic Laws
• For any positive alpha
• Semicircle law
• MP Law
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Additive Free Convolution
• If matrices A,B are asymptotically free, the R-transform of the matrix sum equals the sum of the R-transforms:
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Sigma transform: Definition
• For -1<x<0 (eta definition)
• Reminder: Eta definition
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Sigma transform: Properties & Basic Laws
• AB non-negative definite
• MP law
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Multiplicative Free Convolution
• If matrices A,B are asymptotically free, the Sigma-transform of the matrix product equals the product of the Sigma-transforms:
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Transform Interconnections
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R Stieltjes
(G) Capacity
MMSE η Sigma (S)
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Examples
• Spectrum sensing
– Addition of Wishart matrix functions
• Relay channel, Cochannel interference channel
– Product of Wishart matrix functions
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Spectrum Sensing
• Matrix I/O model
• Covariance of received signal
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Spectrum Sensing
• Pdf trough inversion formula
• Application: SNR estimation
– Measure max eigenvalue from received signal
– Compare to analytic pdf
– Recover SNR p
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Spectrum Sensing
• Sum of
– Standard Wishart matrix
– Scaled Wishart matrix
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Spectrum Sensing
• Stieltjes transform
• Cubic polynomial
– β dimension ratio
– p SNR
– z Stieltjes argument
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Relay Channel
• Vector I/O model
• Mutual information
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Relay Channel
• Asymptotically
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Relay channel
• Product of
– Standard Wishart
– Scaled Wishart plus Identity
• Auxiliary variables
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Relay Channel
• Multiplicative Free Convolution
• Eta transform of M
– Change of variables in MP law
– Eta definition
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Relay Channel
• Quartic polynomial for Stieltjes transform and inversion formula
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Relay Channel
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Cochannel Interference
• Vector I/O model
• Mutual information
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Cochannel Interference
• Product of
– Standard Wishart
– Inverse of Scaled Wishart plus Identity
• Auxiliary variables
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Cochannel Interference
• Asymptotically
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Cochannel Interference
• Multiplicative Free Convolution
• Eta transform of M
– Change of variables in MP law
– Eta definition
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Cochannel Interference
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Summary
• Free probability is a generalization of independence for random matrices
• First, we have to establish that two matrices are asymptotically free
• Matrix sums can be tackled through additive free convolution in R-transform domain
• Matrix products can be tackled through multiplicative free convolution in Sigma-transform domain
• Applications in: – Spectrum Sensing, SNR Estimation
– Cochannel interference, relay channels
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• Questions?
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Random Matrix Theory Lecture 3 Free Probability Theory