Post on 13-Jul-2020
Insights into Large Complex Systemsvia Random Matrix Theory
Lu Wei
SEAS, Harvard
06/23/2016 @UTC Institute for Advanced Systems Engineering
University of Connecticut
Lu Wei Random Matrix Theory and Large Complex Systems 1 / 19
Random Matrix Theory
Random Matrix Theory
Lu Wei Random Matrix Theory and Large Complex Systems 2 / 19
Random Matrix Theory
Random Matrix Theory
milestones
Wishart distribution (Wishart [1928])Semicircle law (Wigner [1955])Marchenko-Pastur law (Marchenko-Pastur [1967])Tracy-Widom law (Tracy-Widom [1990s])KPZ universality class (Johansson [2000s])
tools from almost all branches of mathematics and physicsAkemann et al. (eds) [2011] The Oxford Handbook of Random MatrixTheory. Oxford University Press
applications: biology, data sciences, economics, information theory,machine learning, wireless communications,. . .
Lu Wei Random Matrix Theory and Large Complex Systems 3 / 19
Random Matrix Theory
Random Matrix Theory
milestonesWishart distribution (Wishart [1928])
Semicircle law (Wigner [1955])Marchenko-Pastur law (Marchenko-Pastur [1967])Tracy-Widom law (Tracy-Widom [1990s])KPZ universality class (Johansson [2000s])
tools from almost all branches of mathematics and physicsAkemann et al. (eds) [2011] The Oxford Handbook of Random MatrixTheory. Oxford University Press
applications: biology, data sciences, economics, information theory,machine learning, wireless communications,. . .
Lu Wei Random Matrix Theory and Large Complex Systems 3 / 19
Random Matrix Theory
Random Matrix Theory
milestonesWishart distribution (Wishart [1928])Semicircle law (Wigner [1955])
Marchenko-Pastur law (Marchenko-Pastur [1967])Tracy-Widom law (Tracy-Widom [1990s])KPZ universality class (Johansson [2000s])
tools from almost all branches of mathematics and physicsAkemann et al. (eds) [2011] The Oxford Handbook of Random MatrixTheory. Oxford University Press
applications: biology, data sciences, economics, information theory,machine learning, wireless communications,. . .
Lu Wei Random Matrix Theory and Large Complex Systems 3 / 19
Random Matrix Theory
Random Matrix Theory
milestonesWishart distribution (Wishart [1928])Semicircle law (Wigner [1955])Marchenko-Pastur law (Marchenko-Pastur [1967])
Tracy-Widom law (Tracy-Widom [1990s])KPZ universality class (Johansson [2000s])
tools from almost all branches of mathematics and physicsAkemann et al. (eds) [2011] The Oxford Handbook of Random MatrixTheory. Oxford University Press
applications: biology, data sciences, economics, information theory,machine learning, wireless communications,. . .
Lu Wei Random Matrix Theory and Large Complex Systems 3 / 19
Random Matrix Theory
Random Matrix Theory
milestonesWishart distribution (Wishart [1928])Semicircle law (Wigner [1955])Marchenko-Pastur law (Marchenko-Pastur [1967])Tracy-Widom law (Tracy-Widom [1990s])
KPZ universality class (Johansson [2000s])
tools from almost all branches of mathematics and physicsAkemann et al. (eds) [2011] The Oxford Handbook of Random MatrixTheory. Oxford University Press
applications: biology, data sciences, economics, information theory,machine learning, wireless communications,. . .
Lu Wei Random Matrix Theory and Large Complex Systems 3 / 19
Random Matrix Theory
Random Matrix Theory
milestonesWishart distribution (Wishart [1928])Semicircle law (Wigner [1955])Marchenko-Pastur law (Marchenko-Pastur [1967])Tracy-Widom law (Tracy-Widom [1990s])KPZ universality class (Johansson [2000s])
tools from almost all branches of mathematics and physicsAkemann et al. (eds) [2011] The Oxford Handbook of Random MatrixTheory. Oxford University Press
applications: biology, data sciences, economics, information theory,machine learning, wireless communications,. . .
Lu Wei Random Matrix Theory and Large Complex Systems 3 / 19
Random Matrix Theory
Random Matrix Theory
milestonesWishart distribution (Wishart [1928])Semicircle law (Wigner [1955])Marchenko-Pastur law (Marchenko-Pastur [1967])Tracy-Widom law (Tracy-Widom [1990s])KPZ universality class (Johansson [2000s])
tools from almost all branches of mathematics and physics
Akemann et al. (eds) [2011] The Oxford Handbook of Random MatrixTheory. Oxford University Press
applications: biology, data sciences, economics, information theory,machine learning, wireless communications,. . .
Lu Wei Random Matrix Theory and Large Complex Systems 3 / 19
Random Matrix Theory
Random Matrix Theory
milestonesWishart distribution (Wishart [1928])Semicircle law (Wigner [1955])Marchenko-Pastur law (Marchenko-Pastur [1967])Tracy-Widom law (Tracy-Widom [1990s])KPZ universality class (Johansson [2000s])
tools from almost all branches of mathematics and physicsAkemann et al. (eds) [2011] The Oxford Handbook of Random MatrixTheory. Oxford University Press
applications: biology, data sciences, economics, information theory,machine learning, wireless communications,. . .
Lu Wei Random Matrix Theory and Large Complex Systems 3 / 19
Random Matrix Theory
Random Matrix Theory
milestonesWishart distribution (Wishart [1928])Semicircle law (Wigner [1955])Marchenko-Pastur law (Marchenko-Pastur [1967])Tracy-Widom law (Tracy-Widom [1990s])KPZ universality class (Johansson [2000s])
tools from almost all branches of mathematics and physicsAkemann et al. (eds) [2011] The Oxford Handbook of Random MatrixTheory. Oxford University Press
applications: biology, data sciences, economics, information theory,machine learning, wireless communications,. . .
Lu Wei Random Matrix Theory and Large Complex Systems 3 / 19
Product of Random Matrices
Product of Random Matrices
joint works with Akemann, Hero, Kieburg, Liu, Tarokh, Zhang, Zheng
Lu Wei Random Matrix Theory and Large Complex Systems 4 / 19
Product of Random Matrices
Products of Random Matrices
Crisanti et al. [1993] Products of Random Matrices in StatisticalPhysics. Springer
Müller [2002] On the asymptotic eigenvalue distribution ofconcatenated vector-valued fading channels, IEEE Trans. Inf. Theory
y = Hx + w, where H = Hn · · ·H2H1
H1 H2 Hn
Tx. Rx. cluster 1 cluster 2 cluster n-1
x y
w
an earlier attempt via eigenvalues and singular values relation
Lu Wei Random Matrix Theory and Large Complex Systems 5 / 19
Product of Random Matrices
Products of Random Matrices
Crisanti et al. [1993] Products of Random Matrices in StatisticalPhysics. Springer
Müller [2002] On the asymptotic eigenvalue distribution ofconcatenated vector-valued fading channels, IEEE Trans. Inf. Theory
y = Hx + w, where H = Hn · · ·H2H1
H1 H2 Hn
Tx. Rx. cluster 1 cluster 2 cluster n-1
x y
w
an earlier attempt via eigenvalues and singular values relation
Lu Wei Random Matrix Theory and Large Complex Systems 5 / 19
Product of Random Matrices
Products of Random Matrices
Crisanti et al. [1993] Products of Random Matrices in StatisticalPhysics. Springer
Müller [2002] On the asymptotic eigenvalue distribution ofconcatenated vector-valued fading channels, IEEE Trans. Inf. Theory
y = Hx + w, where H = Hn · · ·H2H1
H1 H2 Hn
Tx. Rx. cluster 1 cluster 2 cluster n-1
x y
w
an earlier attempt via eigenvalues and singular values relation
Lu Wei Random Matrix Theory and Large Complex Systems 5 / 19
Product of Random Matrices
Products of Random Matrices
Crisanti et al. [1993] Products of Random Matrices in StatisticalPhysics. Springer
Müller [2002] On the asymptotic eigenvalue distribution ofconcatenated vector-valued fading channels, IEEE Trans. Inf. Theory
y = Hx + w, where H = Hn · · ·H2H1
H1 H2 Hn
Tx. Rx. cluster 1 cluster 2 cluster n-1
x y
w
an earlier attempt via eigenvalues and singular values relation
Lu Wei Random Matrix Theory and Large Complex Systems 5 / 19
Product of Random Matrices
Products of Random Matrices
Crisanti et al. [1993] Products of Random Matrices in StatisticalPhysics. Springer
Müller [2002] On the asymptotic eigenvalue distribution ofconcatenated vector-valued fading channels, IEEE Trans. Inf. Theory
y = Hx + w, where H = Hn · · ·H2H1
H1 H2 Hn
Tx. Rx. cluster 1 cluster 2 cluster n-1
x y
w
an earlier attempt via eigenvalues and singular values relation
Lu Wei Random Matrix Theory and Large Complex Systems 5 / 19
Product of Random Matrices
Products of Random Matrices
Crisanti et al. [1993] Products of Random Matrices in StatisticalPhysics. Springer
Müller [2002] On the asymptotic eigenvalue distribution ofconcatenated vector-valued fading channels, IEEE Trans. Inf. Theory
y = Hx + w, where H = Hn · · ·H2H1
H1 H2 Hn
Tx. Rx. cluster 1 cluster 2 cluster n-1
x y
w
an earlier attempt via eigenvalues and singular values relation∗
∗W., Zheng, Tirkkonen, Hämäläinen [2013] On the ergodic mutual information of multiple cluster scattering MIMOchannels, IEEE Commun. Lett.
Lu Wei Random Matrix Theory and Large Complex Systems 5 / 19
Product of Random Matrices
Products of Random Matrices
H = Hn · · ·H2H1
n = 1 – Wishart-Laguerre ensemble Bronk [1965]
p (λ1, . . . , λm) ∝ det2(λj−1
k
) m∏i=1
e−λi
n arbitraryAkemann, Kieburg, W. [2013] Singular value correlationfunctions for products of Wishart random matrices, J. Phys. A
p (λ1, . . . , λm) ∝ det(λj−1
k
)det (fj(λk ))
fj(x) = Gm,00,m
(x
∣∣∣∣∣ −0, . . . , 0, j − 1
)= 1
2πı
∮L du x−uΓm−1(u)Γ(u + j − 1)
Lu Wei Random Matrix Theory and Large Complex Systems 6 / 19
Product of Random Matrices
Products of Random Matrices
H = Hn · · ·H2H1
n = 1 – Wishart-Laguerre ensemble Bronk [1965]
p (λ1, . . . , λm) ∝ det2(λj−1
k
) m∏i=1
e−λi
n arbitraryAkemann, Kieburg, W. [2013] Singular value correlationfunctions for products of Wishart random matrices, J. Phys. A
p (λ1, . . . , λm) ∝ det(λj−1
k
)det (fj(λk ))
fj(x) = Gm,00,m
(x
∣∣∣∣∣ −0, . . . , 0, j − 1
)= 1
2πı
∮L du x−uΓm−1(u)Γ(u + j − 1)
Lu Wei Random Matrix Theory and Large Complex Systems 6 / 19
Product of Random Matrices
Products of Random Matrices
H = Hn · · ·H2H1
n = 1 – Wishart-Laguerre ensemble Bronk [1965]
p (λ1, . . . , λm) ∝ det2(λj−1
k
) m∏i=1
e−λi
n arbitraryAkemann, Kieburg, W. [2013] Singular value correlationfunctions for products of Wishart random matrices, J. Phys. A
p (λ1, . . . , λm) ∝ det(λj−1
k
)det (fj(λk ))
fj(x) = Gm,00,m
(x
∣∣∣∣∣ −0, . . . , 0, j − 1
)= 1
2πı
∮L du x−uΓm−1(u)Γ(u + j − 1)
Lu Wei Random Matrix Theory and Large Complex Systems 6 / 19
Product of Random Matrices
Products of Random Matrices
H = Hn · · ·H2H1
n = 1 – Wishart-Laguerre ensemble Bronk [1965]
p (λ1, . . . , λm) ∝ det2(λj−1
k
) m∏i=1
e−λi
n arbitrary†
p (λ1, . . . , λm) ∝ det(λj−1
k
)det (fj(λk ))
fj(x) = Gm,00,m
(x
∣∣∣∣∣ −0, . . . , 0, j − 1
)= 1
2πı
∮L du x−uΓm−1(u)Γ(u + j − 1)
†Akemann, Kieburg, W. [2013] Singular value correlation functions for products of Wishart random matrices, J. Phys. A
Lu Wei Random Matrix Theory and Large Complex Systems 6 / 19
Product of Random Matrices
Products of Random Matrices
H = Hn · · ·H2H1
n = 1 – Wishart-Laguerre ensemble Bronk [1965]
p (λ1, . . . , λm) ∝ det2(λj−1
k
) m∏i=1
e−λi
n arbitrary†
p (λ1, . . . , λm) ∝ det(λj−1
k
)det (fj(λk ))
fj(x) = Gm,00,m
(x
∣∣∣∣∣ −0, . . . , 0, j − 1
)= 1
2πı
∮L du x−uΓm−1(u)Γ(u + j − 1)
†Akemann, Kieburg, W. [2013] Singular value correlation functions for products of Wishart random matrices, J. Phys. A
Lu Wei Random Matrix Theory and Large Complex Systems 6 / 19
Product of Random Matrices
Products of Random Matrices
H = Hn · · ·H2H1
n = 1 – Wishart-Laguerre ensemble Bronk [1965]
p (λ1, . . . , λm) ∝ det2(λj−1
k
) m∏i=1
e−λi
n arbitrary†
p (λ1, . . . , λm) ∝ det(λj−1
k
)det (fj(λk ))
fj(x) = Gm,00,m
(x
∣∣∣∣∣ −0, . . . , 0, j − 1
)= 1
2πı
∮L du x−uΓm−1(u)Γ(u + j − 1)
†Akemann, Kieburg, W. [2013] Singular value correlation functions for products of Wishart random matrices, J. Phys. A
Lu Wei Random Matrix Theory and Large Complex Systems 6 / 19
Product of Random Matrices
Applications to Capacity Analysis
ergodic capacity Akemann-Kieburg-W. [2013]
E[ m∑
i=1
log (1 + γλi)
]
outage capacity of orthogonal space-time codes, i.e., distribution ofm∑
i=1
λi
outage capacity of double-cluster channels, i.e., distribution ofm∑
i=1
log (1 + γλi) for n = 2
Lu Wei Random Matrix Theory and Large Complex Systems 7 / 19
Product of Random Matrices
Applications to Capacity Analysis
ergodic capacity Akemann-Kieburg-W. [2013]
E[ m∑
i=1
log (1 + γλi)
]
outage capacity of orthogonal space-time codes, i.e., distribution ofm∑
i=1
λi
outage capacity of double-cluster channels, i.e., distribution ofm∑
i=1
log (1 + γλi) for n = 2
Lu Wei Random Matrix Theory and Large Complex Systems 7 / 19
Product of Random Matrices
Applications to Capacity Analysis
ergodic capacity Akemann-Kieburg-W. [2013]
E[ m∑
i=1
log (1 + γλi)
]
outage capacity of orthogonal space-time codes‡, i.e., distribution ofm∑
i=1
λi
outage capacity of double-cluster channels, i.e., distribution ofm∑
i=1
log (1 + γλi) for n = 2
‡W., Zheng, Corander, Taricco [2015] On the outage capacity of orthogonal space-time block codes over multi-clusterscattering MIMO channels, IEEE Trans. Commun.
Lu Wei Random Matrix Theory and Large Complex Systems 7 / 19
Product of Random Matrices
Applications to Capacity Analysis
ergodic capacity Akemann-Kieburg-W. [2013]
E[ m∑
i=1
log (1 + γλi)
]
outage capacity of orthogonal space-time codes‡, i.e., distribution ofm∑
i=1
λi
outage capacity of double-cluster channels§, i.e., distribution ofm∑
i=1
log (1 + γλi) for n = 2
‡W., Zheng, Corander, Taricco [2015] On the outage capacity of orthogonal space-time block codes over multi-clusterscattering MIMO channels, IEEE Trans. Commun.
§Zheng, W., Speicher, Müller, Hämäläinen, Corander On the fluctuation of mutual information of double-clusterscattering MIMO channels: A free probability approach, IEEE Trans. Inf. Theory, under revision, arXiv:1502.05516
Lu Wei Random Matrix Theory and Large Complex Systems 7 / 19
Product of Random Matrices
Spiked Products of Random Matrices: Phase Transitions
H = Σ1/2Hn · · ·H2H1
natural generalization E[HH†
]∝ Σ = diag (σ1, . . . , σm) – spikes
n = 1 – Baik-Ben Arous-Péché [2005] Phase transition of the largesteigenvalue for nonnull complex sample covariance matrices, Ann.Probab.
λ1 from Tracy-Widom to Gaussian (BBP phase transition)critical value σcrit = 2
n arbitraryLiu, W., Zhang Singular values for spiked products ofcomplex Ginibre matrices
λ1 BBP phase transitioncritical value σcrit = n + 1
Lu Wei Random Matrix Theory and Large Complex Systems 8 / 19
Product of Random Matrices
Spiked Products of Random Matrices: Phase Transitions
H = Σ1/2Hn · · ·H2H1
natural generalization E[HH†
]∝ Σ = diag (σ1, . . . , σm) – spikes
n = 1 – Baik-Ben Arous-Péché [2005] Phase transition of the largesteigenvalue for nonnull complex sample covariance matrices, Ann.Probab.
λ1 from Tracy-Widom to Gaussian (BBP phase transition)critical value σcrit = 2
n arbitraryLiu, W., Zhang Singular values for spiked products ofcomplex Ginibre matrices
λ1 BBP phase transitioncritical value σcrit = n + 1
Lu Wei Random Matrix Theory and Large Complex Systems 8 / 19
Product of Random Matrices
Spiked Products of Random Matrices: Phase Transitions
H = Σ1/2Hn · · ·H2H1
natural generalization E[HH†
]∝ Σ = diag (σ1, . . . , σm) – spikes
n = 1 – Baik-Ben Arous-Péché [2005] Phase transition of the largesteigenvalue for nonnull complex sample covariance matrices, Ann.Probab.
λ1 from Tracy-Widom to Gaussian (BBP phase transition)critical value σcrit = 2
n arbitraryLiu, W., Zhang Singular values for spiked products ofcomplex Ginibre matrices
λ1 BBP phase transitioncritical value σcrit = n + 1
Lu Wei Random Matrix Theory and Large Complex Systems 8 / 19
Product of Random Matrices
Spiked Products of Random Matrices: Phase Transitions
H = Σ1/2Hn · · ·H2H1
natural generalization E[HH†
]∝ Σ = diag (σ1, . . . , σm) – spikes
n = 1 – Baik-Ben Arous-Péché [2005] Phase transition of the largesteigenvalue for nonnull complex sample covariance matrices, Ann.Probab.
λ1 from Tracy-Widom to Gaussian (BBP phase transition)critical value σcrit = 2
n arbitraryLiu, W., Zhang Singular values for spiked products ofcomplex Ginibre matrices
λ1 BBP phase transitioncritical value σcrit = n + 1
Lu Wei Random Matrix Theory and Large Complex Systems 8 / 19
Product of Random Matrices
Spiked Products of Random Matrices: Phase Transitions
H = Σ1/2Hn · · ·H2H1
natural generalization E[HH†
]∝ Σ = diag (σ1, . . . , σm) – spikes
n = 1 – Baik-Ben Arous-Péché [2005] Phase transition of the largesteigenvalue for nonnull complex sample covariance matrices, Ann.Probab.
λ1 from Tracy-Widom to Gaussian (BBP phase transition)
critical value σcrit = 2
n arbitraryLiu, W., Zhang Singular values for spiked products ofcomplex Ginibre matrices
λ1 BBP phase transitioncritical value σcrit = n + 1
Lu Wei Random Matrix Theory and Large Complex Systems 8 / 19
Product of Random Matrices
Spiked Products of Random Matrices: Phase Transitions
H = Σ1/2Hn · · ·H2H1
natural generalization E[HH†
]∝ Σ = diag (σ1, . . . , σm) – spikes
n = 1 – Baik-Ben Arous-Péché [2005] Phase transition of the largesteigenvalue for nonnull complex sample covariance matrices, Ann.Probab.
λ1 from Tracy-Widom to Gaussian (BBP phase transition)critical value σcrit = 2
n arbitraryLiu, W., Zhang Singular values for spiked products ofcomplex Ginibre matrices
λ1 BBP phase transitioncritical value σcrit = n + 1
Lu Wei Random Matrix Theory and Large Complex Systems 8 / 19
Product of Random Matrices
Spiked Products of Random Matrices: Phase Transitions
H = Σ1/2Hn · · ·H2H1
natural generalization E[HH†
]∝ Σ = diag (σ1, . . . , σm) – spikes
n = 1 – Baik-Ben Arous-Péché [2005] Phase transition of the largesteigenvalue for nonnull complex sample covariance matrices, Ann.Probab.
λ1 from Tracy-Widom to Gaussian (BBP phase transition)critical value σcrit = 2
n arbitrary¶
λ1 BBP phase transitioncritical value σcrit = n + 1
¶Liu, W., Zhang Singular values for spiked products of complex Ginibre matrices
Lu Wei Random Matrix Theory and Large Complex Systems 8 / 19
Product of Random Matrices
Spiked Products of Random Matrices: Phase Transitions
H = Σ1/2Hn · · ·H2H1
natural generalization E[HH†
]∝ Σ = diag (σ1, . . . , σm) – spikes
n = 1 – Baik-Ben Arous-Péché [2005] Phase transition of the largesteigenvalue for nonnull complex sample covariance matrices, Ann.Probab.
λ1 from Tracy-Widom to Gaussian (BBP phase transition)critical value σcrit = 2
n arbitrary¶
λ1 BBP phase transition
critical value σcrit = n + 1
¶Liu, W., Zhang Singular values for spiked products of complex Ginibre matrices
Lu Wei Random Matrix Theory and Large Complex Systems 8 / 19
Product of Random Matrices
Spiked Products of Random Matrices: Phase Transitions
H = Σ1/2Hn · · ·H2H1
natural generalization E[HH†
]∝ Σ = diag (σ1, . . . , σm) – spikes
n = 1 – Baik-Ben Arous-Péché [2005] Phase transition of the largesteigenvalue for nonnull complex sample covariance matrices, Ann.Probab.
λ1 from Tracy-Widom to Gaussian (BBP phase transition)critical value σcrit = 2
n arbitrary¶
λ1 BBP phase transitioncritical value σcrit = n + 1
¶Liu, W., Zhang Singular values for spiked products of complex Ginibre matrices
Lu Wei Random Matrix Theory and Large Complex Systems 8 / 19
Product of Random Matrices
Phase Transitions
50
40
30
λ1
20
10
020
15
10
σ1
5
0.1
0.2
0.3
0
0.4
dens
ity o
f λ
1
MIMO radar detection; community detection,. . .
Lu Wei Random Matrix Theory and Large Complex Systems 9 / 19
Product of Random Matrices
Phase Transitions
50
40
30
λ1
20
10
020
15
10
σ1
5
0.1
0.2
0.3
0
0.4
dens
ity o
f λ
1
MIMO radar detection‖;
community detection,. . .
‖W., Zheng, Hero, Tarokh Scaling laws and phase transitions for target detection in MIMO radar, ITW’16
Lu Wei Random Matrix Theory and Large Complex Systems 9 / 19
Product of Random Matrices
Phase Transitions
50
40
30
λ1
20
10
020
15
10
σ1
5
0.1
0.2
0.3
0
0.4
dens
ity o
f λ
1
MIMO radar detection‖; community detection,. . .
‖W., Zheng, Hero, Tarokh Scaling laws and phase transitions for target detection in MIMO radar, ITW’16
Lu Wei Random Matrix Theory and Large Complex Systems 9 / 19
Other Large Complex Systems
Applications to Other Large Complex Systems
Lu Wei Random Matrix Theory and Large Complex Systems 10 / 19
Signal Processing
Signal Processing
joint works with Dharmawansa, Liang, McKay, Tirkkonen
Lu Wei Random Matrix Theory and Large Complex Systems 11 / 19
Signal Processing
Signal Detection
cognitive radio - a solution to spectrum underutilization problem bydynamic spectrum access
unlicensed users are allowed to opportunistically use the frequencybands that are not heavily occupied by licensed users
awareness of spectrum usage information via spectrum sensing
ym×1
= Hm×p
x + w
Ym×N
= (y1, y2, . . . , yN) =⇒{H0 : noise
H1 : signal + noise
R = YY† data sample covariance; E noise sample covariance
Lu Wei Random Matrix Theory and Large Complex Systems 12 / 19
Signal Processing
Signal Detection
cognitive radio - a solution to spectrum underutilization problem bydynamic spectrum access
unlicensed users are allowed to opportunistically use the frequencybands that are not heavily occupied by licensed users
awareness of spectrum usage information via spectrum sensing
ym×1
= Hm×p
x + w
Ym×N
= (y1, y2, . . . , yN) =⇒{H0 : noise
H1 : signal + noise
R = YY† data sample covariance; E noise sample covariance
Lu Wei Random Matrix Theory and Large Complex Systems 12 / 19
Signal Processing
Signal Detection
cognitive radio - a solution to spectrum underutilization problem bydynamic spectrum access
unlicensed users are allowed to opportunistically use the frequencybands that are not heavily occupied by licensed users
awareness of spectrum usage information via spectrum sensing
ym×1
= Hm×p
x + w
Ym×N
= (y1, y2, . . . , yN) =⇒{H0 : noise
H1 : signal + noise
R = YY† data sample covariance; E noise sample covariance
Lu Wei Random Matrix Theory and Large Complex Systems 12 / 19
Signal Processing
Signal Detection
cognitive radio - a solution to spectrum underutilization problem bydynamic spectrum access
unlicensed users are allowed to opportunistically use the frequencybands that are not heavily occupied by licensed users
awareness of spectrum usage information via spectrum sensing
ym×1
= Hm×p
x + w
Ym×N
= (y1, y2, . . . , yN) =⇒{H0 : noise
H1 : signal + noise
R = YY† data sample covariance; E noise sample covariance
Lu Wei Random Matrix Theory and Large Complex Systems 12 / 19
Signal Processing
Signal Detection
cognitive radio - a solution to spectrum underutilization problem bydynamic spectrum access
unlicensed users are allowed to opportunistically use the frequencybands that are not heavily occupied by licensed users
awareness of spectrum usage information via spectrum sensing
ym×1
= Hm×p
x + w
Ym×N
= (y1, y2, . . . , yN) =⇒{H0 : noise
H1 : signal + noise
R = YY† data sample covariance; E noise sample covariance
Lu Wei Random Matrix Theory and Large Complex Systems 12 / 19
Signal Processing
Signal Detection
cognitive radio - a solution to spectrum underutilization problem bydynamic spectrum access
unlicensed users are allowed to opportunistically use the frequencybands that are not heavily occupied by licensed users
awareness of spectrum usage information via spectrum sensing
ym×1
= Hm×p
x + w
Ym×N
= (y1, y2, . . . , yN) =⇒{H0 : noise
H1 : signal + noise
R = YY† data sample covariance; E noise sample covariance
Lu Wei Random Matrix Theory and Large Complex Systems 12 / 19
Signal Processing
Signal Detection
p > 1 but unknown∗
TST =det (R)(1m tr(R)
)m
low signal-to-noise ratio†
TJ =tr(R2)
tr2(R)
unknown noise covariance‡
TW =det (E)
det (R + E)
∗W., Tirkkonen [2012] Spectrum sensing in the presence of multiple primary users, IEEE Trans. Commun.
†W., Dharmawansa, Tirkkonen [2013] Multiple primary user spectrum sensing in the low SNR regime, IEEE Trans.Commun.‡W., Tirkkonen, Liang [2014] Multi-source signal detection with arbitrary noise covariance, IEEE Trans. Signal Process.
Lu Wei Random Matrix Theory and Large Complex Systems 13 / 19
Signal Processing
Signal Detection
p > 1 but unknown∗
TST =det (R)(1m tr(R)
)m
low signal-to-noise ratio†
TJ =tr(R2)
tr2(R)
unknown noise covariance‡
TW =det (E)
det (R + E)
∗W., Tirkkonen [2012] Spectrum sensing in the presence of multiple primary users, IEEE Trans. Commun.†W., Dharmawansa, Tirkkonen [2013] Multiple primary user spectrum sensing in the low SNR regime, IEEE Trans.
Commun.
‡W., Tirkkonen, Liang [2014] Multi-source signal detection with arbitrary noise covariance, IEEE Trans. Signal Process.
Lu Wei Random Matrix Theory and Large Complex Systems 13 / 19
Signal Processing
Signal Detection
p > 1 but unknown∗
TST =det (R)(1m tr(R)
)m
low signal-to-noise ratio†
TJ =tr(R2)
tr2(R)
unknown noise covariance‡
TW =det (E)
det (R + E)
∗W., Tirkkonen [2012] Spectrum sensing in the presence of multiple primary users, IEEE Trans. Commun.†W., Dharmawansa, Tirkkonen [2013] Multiple primary user spectrum sensing in the low SNR regime, IEEE Trans.
Commun.‡W., Tirkkonen, Liang [2014] Multi-source signal detection with arbitrary noise covariance, IEEE Trans. Signal Process.
Lu Wei Random Matrix Theory and Large Complex Systems 13 / 19
Coding Theory
Coding Theory
joint works with Corander, Pitaval, Tirkkonen
Lu Wei Random Matrix Theory and Large Complex Systems 14 / 19
Coding Theory
Coding Theory
fundamental issue: cardinality and minimum distance tradeoff
a code with cardinality |C|, C ={
G1,G2, . . . ,G|C|}⊂ G
minimum distance, r = min{||Gi − Gj ||
∣∣ Gi ,Gj ∈ C, i 6= j}
1µ (B (r))︸ ︷︷ ︸
Gilbert-Varshamov bound
≤ |C| ≤ 1µ (B (r/2))︸ ︷︷ ︸
Hamming bound
metric ball, B (r) ={
G ∈ G∣∣ d(
G,G′)≤ r
}, G
′ ∈ Gvolume of metric ball, µ (B (r)) =
∫d(G,G′)≤r f (G) dG
Lu Wei Random Matrix Theory and Large Complex Systems 15 / 19
Coding Theory
Coding Theory
fundamental issue: cardinality and minimum distance tradeoff
a code with cardinality |C|, C ={
G1,G2, . . . ,G|C|}⊂ G
minimum distance, r = min{||Gi − Gj ||
∣∣ Gi ,Gj ∈ C, i 6= j}
1µ (B (r))︸ ︷︷ ︸
Gilbert-Varshamov bound
≤ |C| ≤ 1µ (B (r/2))︸ ︷︷ ︸
Hamming bound
metric ball, B (r) ={
G ∈ G∣∣ d(
G,G′)≤ r
}, G
′ ∈ Gvolume of metric ball, µ (B (r)) =
∫d(G,G′)≤r f (G) dG
Lu Wei Random Matrix Theory and Large Complex Systems 15 / 19
Coding Theory
Coding Theory
fundamental issue: cardinality and minimum distance tradeoff
a code with cardinality |C|, C ={
G1,G2, . . . ,G|C|}⊂ G
minimum distance, r = min{||Gi − Gj ||
∣∣ Gi ,Gj ∈ C, i 6= j}
1µ (B (r))︸ ︷︷ ︸
Gilbert-Varshamov bound
≤ |C| ≤ 1µ (B (r/2))︸ ︷︷ ︸
Hamming bound
metric ball, B (r) ={
G ∈ G∣∣ d(
G,G′)≤ r
}, G
′ ∈ Gvolume of metric ball, µ (B (r)) =
∫d(G,G′)≤r f (G) dG
Lu Wei Random Matrix Theory and Large Complex Systems 15 / 19
Coding Theory
Coding Theory
fundamental issue: cardinality and minimum distance tradeoff
a code with cardinality |C|, C ={
G1,G2, . . . ,G|C|}⊂ G
minimum distance, r = min{||Gi − Gj ||
∣∣ Gi ,Gj ∈ C, i 6= j}
1µ (B (r))︸ ︷︷ ︸
Gilbert-Varshamov bound
≤ |C| ≤ 1µ (B (r/2))︸ ︷︷ ︸
Hamming bound
metric ball, B (r) ={
G ∈ G∣∣ d(
G,G′)≤ r
}, G
′ ∈ Gvolume of metric ball, µ (B (r)) =
∫d(G,G′)≤r f (G) dG
Lu Wei Random Matrix Theory and Large Complex Systems 15 / 19
Coding Theory
Coding Theory
fundamental issue: cardinality and minimum distance tradeoff
a code with cardinality |C|, C ={
G1,G2, . . . ,G|C|}⊂ G
minimum distance, r = min{||Gi − Gj ||
∣∣ Gi ,Gj ∈ C, i 6= j}
1µ (B (r))︸ ︷︷ ︸
Gilbert-Varshamov bound
≤ |C| ≤ 1µ (B (r/2))︸ ︷︷ ︸
Hamming bound
metric ball, B (r) ={
G ∈ G∣∣ d(
G,G′)≤ r
}, G
′ ∈ G
volume of metric ball, µ (B (r)) =∫
d(G,G′)≤r f (G) dG
Lu Wei Random Matrix Theory and Large Complex Systems 15 / 19
Coding Theory
Coding Theory
fundamental issue: cardinality and minimum distance tradeoff
a code with cardinality |C|, C ={
G1,G2, . . . ,G|C|}⊂ G
minimum distance, r = min{||Gi − Gj ||
∣∣ Gi ,Gj ∈ C, i 6= j}
1µ (B (r))︸ ︷︷ ︸
Gilbert-Varshamov bound
≤ |C| ≤ 1µ (B (r/2))︸ ︷︷ ︸
Hamming bound
metric ball, B (r) ={
G ∈ G∣∣ d(
G,G′)≤ r
}, G
′ ∈ Gvolume of metric ball, µ (B (r)) =
∫d(G,G′)≤r f (G) dG
Lu Wei Random Matrix Theory and Large Complex Systems 15 / 19
Coding Theory
Volume of Metric Balls
Lu Wei Random Matrix Theory and Large Complex Systems 16 / 19
Coding Theory
Unitary Group
µ (B (r)) ∝∫· · ·∫||U−In||F≤r
∏1≤j<k≤n
∣∣∣eıθj − eıθk∣∣∣2 n∏
i=1
dθi
Han-Rosenthal [2006] Unitary space-time constellation analysis: anupper bound for the diversity, IEEE Trans. Inf. Theory
limiting behaviorW., Pitaval, Corander, Tirkkonen From random matrixtheory to coding theory: Volume of a metric ball in unitary group,IEEE Trans. Inf. Theory, submitted, arXiv:1506.07259 as n→∞
µ (B (r)) ' 12
erf(n)− 12
erf
(n − r2
2
)
super-exponential rate of convergence O (n−cn)CLT of linear statistics of unitary group
Lu Wei Random Matrix Theory and Large Complex Systems 17 / 19
Coding Theory
Unitary Group
µ (B (r)) ∝∫· · ·∫||U−In||F≤r
∏1≤j<k≤n
∣∣∣eıθj − eıθk∣∣∣2 n∏
i=1
dθi
Han-Rosenthal [2006] Unitary space-time constellation analysis: anupper bound for the diversity, IEEE Trans. Inf. Theory
limiting behaviorW., Pitaval, Corander, Tirkkonen From random matrixtheory to coding theory: Volume of a metric ball in unitary group,IEEE Trans. Inf. Theory, submitted, arXiv:1506.07259 as n→∞
µ (B (r)) ' 12
erf(n)− 12
erf
(n − r2
2
)
super-exponential rate of convergence O (n−cn)CLT of linear statistics of unitary group
Lu Wei Random Matrix Theory and Large Complex Systems 17 / 19
Coding Theory
Unitary Group
µ (B (r)) ∝∫· · ·∫||U−In||F≤r
∏1≤j<k≤n
∣∣∣eıθj − eıθk∣∣∣2 n∏
i=1
dθi
Han-Rosenthal [2006] Unitary space-time constellation analysis: anupper bound for the diversity, IEEE Trans. Inf. Theory
limiting behaviorW., Pitaval, Corander, Tirkkonen From random matrixtheory to coding theory: Volume of a metric ball in unitary group,IEEE Trans. Inf. Theory, submitted, arXiv:1506.07259 as n→∞
µ (B (r)) ' 12
erf(n)− 12
erf
(n − r2
2
)
super-exponential rate of convergence O (n−cn)CLT of linear statistics of unitary group
Lu Wei Random Matrix Theory and Large Complex Systems 17 / 19
Coding Theory
Unitary Group
µ (B (r)) ∝∫· · ·∫||U−In||F≤r
∏1≤j<k≤n
∣∣∣eıθj − eıθk∣∣∣2 n∏
i=1
dθi
Han-Rosenthal [2006] Unitary space-time constellation analysis: anupper bound for the diversity, IEEE Trans. Inf. Theory
limiting behavior∗ as n→∞
µ (B (r)) ' 12
erf(n)− 12
erf
(n − r2
2
)
super-exponential rate of convergence O (n−cn)CLT of linear statistics of unitary group
∗W., Pitaval, Corander, Tirkkonen From random matrix theory to coding theory: Volume of a metric ball in unitary group,IEEE Trans. Inf. Theory, submitted, arXiv:1506.07259
Lu Wei Random Matrix Theory and Large Complex Systems 17 / 19
Coding Theory
Unitary Group
µ (B (r)) ∝∫· · ·∫||U−In||F≤r
∏1≤j<k≤n
∣∣∣eıθj − eıθk∣∣∣2 n∏
i=1
dθi
Han-Rosenthal [2006] Unitary space-time constellation analysis: anupper bound for the diversity, IEEE Trans. Inf. Theory
limiting behavior∗ as n→∞
µ (B (r)) ' 12
erf(n)− 12
erf
(n − r2
2
)
super-exponential rate of convergence O (n−cn)
CLT of linear statistics of unitary group
∗W., Pitaval, Corander, Tirkkonen From random matrix theory to coding theory: Volume of a metric ball in unitary group,IEEE Trans. Inf. Theory, submitted, arXiv:1506.07259
Lu Wei Random Matrix Theory and Large Complex Systems 17 / 19
Coding Theory
Unitary Group
µ (B (r)) ∝∫· · ·∫||U−In||F≤r
∏1≤j<k≤n
∣∣∣eıθj − eıθk∣∣∣2 n∏
i=1
dθi
Han-Rosenthal [2006] Unitary space-time constellation analysis: anupper bound for the diversity, IEEE Trans. Inf. Theory
limiting behavior∗ as n→∞
µ (B (r)) ' 12
erf(n)− 12
erf
(n − r2
2
)
super-exponential rate of convergence O (n−cn)CLT of linear statistics of unitary group
∗W., Pitaval, Corander, Tirkkonen From random matrix theory to coding theory: Volume of a metric ball in unitary group,IEEE Trans. Inf. Theory, submitted, arXiv:1506.07259
Lu Wei Random Matrix Theory and Large Complex Systems 17 / 19
Coding Theory
Grassmann Manifold
µ (B (r)) ∝∫· · ·∫
0≤xi≤1∑pi=1 xi≤r
∏1≤j<k≤p
(xj − xk )2p∏
i=1
xn−p−qi (1− xi)
q−p dxi
Barg-Nogin [2002] fixed p, q, large n; Dai et al. [2008] r ≤ 1
limiting behavior as n, p, q →∞
µ (B (r)) ' 12
erf(
b√2a
)− 1
2erf
(b − r2√
2a
)
CLT of linear statistics of Jacobi ensemblemoments from Painlevé V
Lu Wei Random Matrix Theory and Large Complex Systems 18 / 19
Coding Theory
Grassmann Manifold
µ (B (r)) ∝∫· · ·∫
0≤xi≤1∑pi=1 xi≤r
∏1≤j<k≤p
(xj − xk )2p∏
i=1
xn−p−qi (1− xi)
q−p dxi
Barg-Nogin [2002] fixed p, q, large n; Dai et al. [2008] r ≤ 1
limiting behavior as n, p, q →∞
µ (B (r)) ' 12
erf(
b√2a
)− 1
2erf
(b − r2√
2a
)
CLT of linear statistics of Jacobi ensemblemoments from Painlevé V
Lu Wei Random Matrix Theory and Large Complex Systems 18 / 19
Coding Theory
Grassmann Manifold
µ (B (r)) ∝∫· · ·∫
0≤xi≤1∑pi=1 xi≤r
∏1≤j<k≤p
(xj − xk )2p∏
i=1
xn−p−qi (1− xi)
q−p dxi
Barg-Nogin [2002] fixed p, q, large n; Dai et al. [2008] r ≤ 1
limiting behavior as n, p, q →∞
µ (B (r)) ' 12
erf(
b√2a
)− 1
2erf
(b − r2√
2a
)
CLT of linear statistics of Jacobi ensemblemoments from Painlevé V
Lu Wei Random Matrix Theory and Large Complex Systems 18 / 19
Coding Theory
Grassmann Manifold
µ (B (r)) ∝∫· · ·∫
0≤xi≤1∑pi=1 xi≤r
∏1≤j<k≤p
(xj − xk )2p∏
i=1
xn−p−qi (1− xi)
q−p dxi
Barg-Nogin [2002] fixed p, q, large n; Dai et al. [2008] r ≤ 1
limiting behavior† as n, p, q →∞
µ (B (r)) ' 12
erf(
b√2a
)− 1
2erf
(b − r2√
2a
)
CLT of linear statistics of Jacobi ensemblemoments from Painlevé V
†Pitaval, W., Tirkkonen, Corander Volume of metric balls in high-dimensional complex Grassmann manifolds, IEEETrans. Inf. Theory, submitted, arXiv:1508.00256
Lu Wei Random Matrix Theory and Large Complex Systems 18 / 19
Coding Theory
Grassmann Manifold
µ (B (r)) ∝∫· · ·∫
0≤xi≤1∑pi=1 xi≤r
∏1≤j<k≤p
(xj − xk )2p∏
i=1
xn−p−qi (1− xi)
q−p dxi
Barg-Nogin [2002] fixed p, q, large n; Dai et al. [2008] r ≤ 1
limiting behavior† as n, p, q →∞
µ (B (r)) ' 12
erf(
b√2a
)− 1
2erf
(b − r2√
2a
)
CLT of linear statistics of Jacobi ensemble
moments from Painlevé V
†Pitaval, W., Tirkkonen, Corander Volume of metric balls in high-dimensional complex Grassmann manifolds, IEEETrans. Inf. Theory, submitted, arXiv:1508.00256
Lu Wei Random Matrix Theory and Large Complex Systems 18 / 19
Coding Theory
Grassmann Manifold
µ (B (r)) ∝∫· · ·∫
0≤xi≤1∑pi=1 xi≤r
∏1≤j<k≤p
(xj − xk )2p∏
i=1
xn−p−qi (1− xi)
q−p dxi
Barg-Nogin [2002] fixed p, q, large n; Dai et al. [2008] r ≤ 1
limiting behavior† as n, p, q →∞
µ (B (r)) ' 12
erf(
b√2a
)− 1
2erf
(b − r2√
2a
)
CLT of linear statistics of Jacobi ensemblemoments from Painlevé V
†Pitaval, W., Tirkkonen, Corander Volume of metric balls in high-dimensional complex Grassmann manifolds, IEEETrans. Inf. Theory, submitted, arXiv:1508.00256
Lu Wei Random Matrix Theory and Large Complex Systems 18 / 19
Conclusion
Large Complex Systems
Random Matrix Theory
Wireless Communications
Statistical Physics
Information Theory
Signal Processing
Machine Learning
Data Sciences
?
Lu Wei Random Matrix Theory and Large Complex Systems 19 / 19