Applications of Large Random Matrices to Digital ...whachem/rmt_eusipco.pdfHachem, Loubaton and...
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Applications of Large Random Matrices
to Digital Communicationsand Statistical Signal Processing
W. Hachem, Ph. Loubaton and J. Najim
EUSIPCO 2011Barcelona
August / September 2011
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1 Problem statementIntroduction to the Marcenko-Pastur distribution.Some generalizations.Short review of important previous works.Brief overview of applications to digital communicationsIntroduction to the applications to statistical signal processing
2 K = 0: An overview of Marcenko and Pastur’s results
3 K fixed: spiked models
4 K may scale with M. Application to the subspace method.
5 Some research prospects
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Fundamental example.
V =
V11 V12 . . . V1N
V21 V22 . . . V2N...
......
...VM1 VM2 . . . VMN
(Vij)1≤i≤M,1≤j≤N i.i.d. complex Gaussian random variables CN (0, σ2).v1, v2, . . . , vN columns of V, R = E(vnv
Hn ) = σ2IM
Empirical covariance matrix:
R =1
NVVH =
1
N
N∑
n=1
vnvHn
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Behaviour of the empirical distribution of the eigenvalues of R largeM and N.
How behave the histograms of the eigenvalues (λi )i=1,...,M of R when M
and N increase.
Well known case: M fixed, N increases i.e. MN
small
1N
∑Nn=1 vnv
Hn ≃ E(vnv
Hn ) = σ2IM by the law of large numbers.
If N >> M , the eigenvalues of 1NVVH are concentrated around σ2.
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Illustration.
M = 256, MN
= 1256
, σ2 = 1
−0.5 0 0.5 1 1.5 2 2.5 30
10
20
30
40
50
60
70
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If M et N are of the same order of magnitude.
M,N → +∞ such that MN
= cN ∈ [a, b], a > 0, b < +∞.
Ri ,j ≃ σ2δi−j but
‖R− σ2IM‖ does not converge torwards 0.
The histograms of the eigenvalues of R tend to concentrate aroundthe probability density of the so-called Marcenko-Pastur distribution:
If cN ≤ 1,
pcN (λ) =1
2πcNλ
√[σ2(1 +
√cN)2 − λ][λ− σ2(1−√
cN)2]
if λ ∈ [σ2(1−√cN)
2, σ2(1 +√cN)
2]
= 0 otherwise
Result still true in the non Gaussian case
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Illustrations I.
M = 256, MN
= 116, σ2 = 1
−0.5 0 0.5 1 1.5 2 2.5 30
2
4
6
8
10
12
14
16
18
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Illustrations II.
M = 256, MN
= 14, σ2 = 1
−0.5 0 0.5 1 1.5 2 2.5 30
2
4
6
8
10
12
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Illustrations III.
M = 256, MN
= 2/3, σ2 = 1
−0.5 0 0.5 1 1.5 2 2.5 3 3.5 40
5
10
15
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Possible to evaluate the asymptotic behaviour of linear
statistics
1
M
M∑
k=1
f (λk) =1
MTrace(f (R)) ≃
∫f (λ)pcN (λ) dλ
Example 1: f (λ) = 1ρ2+λ
1MTrace
(R+ ρ2I
)−1≃∫ pcN (λ)
ρ2+σ2 dλ = mN(−ρ2)
mN(−ρ2) unique positive solution of the equation
mN(−ρ2) =1
ρ2 + σ2
1+σ2cNmN (−ρ2)
Closed form solution (see below)
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Example 2: f (λ) = log(1 + λρ2)
1M
log det(IM + R
ρ2
)nearly equal to
1
cNlog(1 + σ2cNmN(−ρ2)
)+ log
(1 + σ2cNmN(−ρ2) + (1 − cN)
σ2
ρ2
)
−ρ2σ2mN(−ρ2)(cNmN(−ρ2) +
1− cN
ρ2
)
Closed form formula
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Fluctuations of the linear statistics.
The bias
E
[1
MTr(f (R)
)]=
∫f (λ) pcN (λ)dλ+O(
1
M2)
The variance
M
[1
MTr(f (R)
)−∫
f (λ) pcN (λ)dλ
]→ N (0,∆2)
In other words:
1
MTr(f (R)
)−∫
f (λ) pcN (λ)dλ ≃ N (0,∆2
M2)
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1 Problem statementIntroduction to the Marcenko-Pastur distribution.Some generalizations.Short review of important previous works.Brief overview of applications to digital communicationsIntroduction to the applications to statistical signal processing
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Generalizations of these behaviours, W = V√N.
Y = C1/2W, C ≥ 0 deterministic, zero mean correlated model.
Y = C1/2WC1/2, C ≥ 0, C ≥ 0 deterministic, zero mean bi-correlatedmodel also known as Kronecker model in the MIMO context.
Y = A+W, A deterministic, information plus noise model.
Y = A+ C1/2WC1/2, Rician bi-correlated MIMO channel.
Y = U (∆⊙W)QH , U,Q unitary deterministic matrices, ∆deterministic, Sayeed model.
Y = A+U (∆⊙W)QH , non zero mean Sayeed model.
Replace i.i.d. matrix W by an isometric random Haar distributedmatrix (obtained from a Gram-Schmidt orthogonalization of W whencN > 1).
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1 Problem statementIntroduction to the Marcenko-Pastur distribution.Some generalizations.Short review of important previous works.Brief overview of applications to digital communicationsIntroduction to the applications to statistical signal processing
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Some important contributors.
In statistical physics.
Wigner (1950), Dyson, Mehta, Brezin, ....
In probability theory
Marcenko, Pastur and colleagues from 1967, Girko from 1975, Bai,Silverstein from 1985.
Voiculescu and the discovery of the free probability theory from 1993.
From 1995, a large community using various techniques.
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In our field
Digital communications: from 1997
Seminal works of Tse and colleagues and Verdu and colleagues in1997 on performance analysis of large CDMA systems
Performance analysis of large MIMO systems
Various applications to ressource allocation
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Statistics and statistical signal processing
Before 2007, some works of Girko who was the first to addressparameter estimation problems in the context of large randommatrices.
El-Karoui (2008) followed by a number of other researchers addressedthe population estimation: estimate the entries of diagonal matrix P
from matrix 1NVPVH .
Seminal works of Mestre-Lagunas (2008) and Mestre (2008) on thebehaviour of the subspace method when the number of sensors andthe number of snapshots converge torward ∞ at the same rate.
More recent works on applications to source number estimation(Nadler 2010), to source detection (Bianchi et al. 2011), to powerdistribution estimation problems in the context of multiuserscommunication systems (Couillet et al. 2011).
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1 Problem statementIntroduction to the Marcenko-Pastur distribution.Some generalizations.Short review of important previous works.Brief overview of applications to digital communicationsIntroduction to the applications to statistical signal processing
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Performance analysis of large CDMA systems .
The simplest context: Tse and Hanly, Verdu and Shamai 1999
M spreading factor, K number of users
received M–dimensional vector y = hWs+ n
s K–dimensional vector of the transmitted symbols
n additive white noise, E(nnH) = ρ2IM
W M × K matrix of the codes allocated to the users, modelled as arealization of a zero mean i.i.d. matrix such that E|Wi ,j |2 = 1
M
h amplitude of the received signal
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Performance of the MMSE receiver.
MMSE Estimation of s1, W = (w1,W2)
SINR βM = wH1
(W2W
H2 + ρ2
|h|2)−1
w1
Analysis of βM when M ,K → ∞, in such a way that MK∈ [a, b]
βM ≃ βM = 1MTr(W2W
H2 + ρ2
|h|2)−1
βM ≃ βM,∗ deterministic positive solution of the equation
βM,∗ =1
ρ2
|h|2 +K−1M
11+βM,∗
Allows to have a better understanding of the MMSE receiver: find theloading factor for which βM,∗ is above a target SINR, find the loadingfactor maximizing the throughput K
Mlog(1 + βM,∗), ...
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Examples of extensions to more realistic models.
Downlink with frequency selective channel (Debbah et.al. 2003)
y = HWs+ n, H Toeplitz matrix
βM,∗ =1
M
M−1∑
m=0
1ρ2
|h(e2iπm/M )|2 +K−1M
11+βM,∗
Downlink with frequency selective channel and random orthogonalHaar distributed code matrix (Debbah et.al. 2003)
βM,∗ =1
M
M−1∑
m=0
1
ρ2
|h(e2iπm/M )|2(1− K−1
M
βM,∗
1+βM,∗
)+ K−1
M1
1+βM,∗
Uplink with frequency selective channel (Li et.al. 2004)
y =∑K
k=1 Hkwksk + n; the channel matrices Hk are Toeplitz.
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Applications to optimal precoding of MIMO systems.
M receive antennas, N transmit antennas
y = Hx + n
H MIMO channel, M × N non observable Gaussian random matrixwith known (or well estimated) second order statistics
x transmitted vector
n additive white Gaussian noise, E(nnH) = ρ2IM
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The optimum precoding problem
Find the covariance matrix Q of x so as to maximize some figure of meritof the system
Typical example: I (Q) = E
[log det
(IM + HQHH
ρ2
)]
To be maximized w.r.t. Q on the convex domain Q ≥ 0 and 1MTr(Q) ≤ 1.
Q → I (Q) is a concave function, but is in general difficult to evaluate inclosed form its gradient and hessian. Have to be evaluated usingMonte-Carlo simulations (Vu-Paulraj 2005).
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A possible alternative: maximize a large system
approximation of I (Q)
Example of bicorrelated Rician channels H = A + C1/2WC1/2
(Dumont et.al. 2010)
Eigenvectors of the optimum matrix Q∗ have no closed formexpression
I (Q) = I (Q) +O( 1M), and I (Q∗) = I (Q∗) +O( 1
M) where
I (Q) = log det(IM +Q× G
(δ(Q), δ(Q)
))+ j(δ(Q), δ(Q)
)
where(δ(Q), δ(Q)
)are the unique solutions of a system of 2 non
linear equations depending on Q,A,C, C,
G is a matrix valued function of(δ(Q), δ(Q)
)given in closed form,
j(δ(Q), δ(Q)
)is a function of
(δ(Q), δ(Q)
)given in closed form.
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Maximization of I (Q) using an iterative waterfilling
algorithm
I (Q) = log det[IM +Q× G
(δ(Q), δ(Q)
)]+ j(δ(Q), δ(Q)
).
Q(k−1) available
Compute(δ(Q(k−1)), δ(Q(k−1))
)= (δ(k−1), δ(k−1))
Q(k) = Argmax log det(IM +Q× G(δ(k−1), δ(k−1))
): waterfilling
k=k+1
If the algorithm converges, it converges torwards Q∗
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1 Problem statementIntroduction to the Marcenko-Pastur distribution.Some generalizations.Short review of important previous works.Brief overview of applications to digital communicationsIntroduction to the applications to statistical signal processing
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The model considered in the following
Observation: M–dimensional time series yn observed fromn = 1, . . . ,N.
yn =∑K
k=1 aksk,n + vn = Asn + vn
((sk,n)n∈Z)k=1,K are K < M non observable ”source signals”,sn = (s1,n, . . . , sK ,n)
T
A = (a1, . . . , aK ) deterministic unknown rank K < M matrix
(vn)n∈Z additive complex white Gaussian noise such thatE(vnv
Hn ) = σ2IM
In matrix form
YN = (y1, . . . , yN) observation M × N matrix
YN = ASN +VN
ΣN = YN√N, BN = A SN√
N, WN = VN√
N
ΣN = BN +WN
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The problems to be addressed.
Detection of the presence of signal(s) from matrix ΣN
K = 1 versus K = 0 to simplify
Various generalizations are possible
Estimation of direction of arrival (DOA) from matrix ΣN .
ak = a(ϕk) where ϕ→ a(ϕ) is known
Estimate the parameters (ϕk)k=1,...,K
Problems addressed when M and N are of the same order of magnitude:M,N → ∞ while the ratio cN = M
Nis bounded away from 0 and upper
bounded.
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In the following
Study of the properties of ΣN when
K = 0, noise only
K does not scale with M, i.e. K ≪ M, spiked model: applicationsto the detection K = 1 versus K = 0, application to the subspaceDOA estimation method
K may scale with M, i.e. K is not much less than 0, application tothe subspace DOA estimation method
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1 Problem statement
2 K = 0: An overview of Marcenko and Pastur’s resultsThe Stieltjes transformGaussian toolsMarcenko-Pastur Probability distributionA symmetric view of Marcenko-Pastur equationBehavior of the individual entries of the resolventFiner convergence results
3 K fixed: spiked models
4 K may scale with M. Application to the subspace method.
5 Some research prospects
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The Stieltjes transform I (measure with density)
The Stieltjes transform is one of the numerous transforms associated toa measure. It is particularly well-suited to study Large Random Matricesand was introduced in this context by Marcenko and Pastur (1967).
Definition
If the measure µ admits a density f with support S:
dµ(λ) = f (λ)dλ on S ,
then the Stieltjes transform Ψµ(z) is defined as:
Ψµ(z) =
∫
S
f (λ)
λ− zdλ ,
= −∞∑
k=0
z−(k+1)
(∫
Sλk f (λ) dλ
)
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The Stieltjes transform I (properties)Let im(z) be the imaginary part of z ∈ C.
Property 1 - identical sign for imaginary part
imΨµ(z) = im(z)
∫
S
f (λ)
(λ− x)2dλ
Property 2 - monotonicity
If z = x ∈ R \ S, then Ψµ(x) well-defined and:
Ψ′µ(x) =
∫
S
f (λ)
(λ− x)2dλ > 0 ⇒ Ψµ(x) ր on R \ S .
Property 3 - Inverse formula
f (λ) =1
πlim
y→0+imΨµ(λ+ ιy) ,
Note that if λ ∈ R \ S, then Ψµ(x) ∈ R ⇒ f (λ) = 0.
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The Stieltjes transform II (measure with Dirac
components)
Stieltjes transform for a Dirac measure
Let δx be the Dirac measure at x : δx(A) =
{1 if x ∈ A,0 else.
Then
Ψδx (z) =1
x − zin particular, Ψδ0(z) = −1
z.
◮ Important example:
LM =1
M
M∑
k=1
δλk⇒ ΨLM (z) =
1
M
M∑
k=1
1
λk − z.
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The Stieltjes transform III (link with the resolvent)
Let X be a M ×M Hermitian matrix:
X = U
λ1 0. . .
0 λM
U∗
and consider its resolvent Q(z) and spectral measure LM :
Q(z) = (X− zI)−1 , LM =1
M
M∑
k=1
δλk.
The Stieltjes transform of the spectral measure is the normalized trace ofthe resolvent:
ΨLM (z) =1
MtrQ(z) .
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2 K = 0: An overview of Marcenko and Pastur’s resultsThe Stieltjes transformGaussian toolsMarcenko-Pastur Probability distributionA symmetric view of Marcenko-Pastur equationBehavior of the individual entries of the resolventFiner convergence results
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Gaussian tools
Let the Zi ’s be independent complex Gaussian random variables anddenote by z = (Z1, · · · ,Zn). The two following results are extremelyefficient when dealing with matrices with Gaussian entries (Pastur 2005).
Integration by part Formula
E (ZkΦ(z, z)) = E|Zk |2E(∂Φ
∂Z k
)
Poincare-Nash Inequality
var (Φ(z, z)) ≤n∑
k=1
E|Zk |2(E
∣∣∣∣∂Φ
∂Zk
∣∣∣∣2
+ E
∣∣∣∣∂Φ
∂Z k
∣∣∣∣2)
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2 K = 0: An overview of Marcenko and Pastur’s resultsThe Stieltjes transformGaussian toolsMarcenko-Pastur Probability distributionA symmetric view of Marcenko-Pastur equationBehavior of the individual entries of the resolventFiner convergence results
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Marcenko - Pastur Probability distribution
We go back to Marcenko and Pastur framework and consider
WN =VN√N
where VN is a M × N matrix with i.i.d. complex Gaussian randomvariables CN (0, σ2).We are interested in the limiting spectral distribution of WNW
∗N . Consider
the associated resolvent and Stieltjes transform:
Q(z) = (WNW∗N − zI)−1 , mN(z) =
1
MtrQ(z) .
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Marcenko - Pastur Probability distribution
We compute hereafter the equation satisfied by the Stieltjes transformassociated to the limiting spectral distribution. Afterwards, we rely on theinverse formula for Stietjes transforms to get Marcenko - Pasturdistribution.
Main assumption
The ratio cN = MN
is bounded away from zero and upper bounded asM,N → ∞.
The three main steps are:
1 To prove that var (mN(z)) = O(N−2). This enables to replace
mN(z) by EmN(z) in the computations .
2 To establish the limiting equation satisfied by EmN(z).
3 To recover the probability distribution with the help of the inverseformula for Stietjes transforms.
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Step 1: Marcenko - Pastur Equation
Proposition
var (mN(z)) = O(
1
N2
).
Proof:∂Qr ,r
∂Wij
= −(Qwj)rQi ,r
By summing over r , then over i and j , we obtain:
∑
i ,j
E
∣∣∣∣∣∂mN(z)
∂Wij
∣∣∣∣∣
2
= E
(1
M2tr Q2WW∗Q2∗
)
≤ 1
| im(z)|4E(
1
M2tr WW∗
)= O
(1
M
).
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Step 1: Marcenko - Pastur Equation (end of proof)
By Poincare-Nash inequality
var (mN(z)) ≤∑
i ,j
E|Wij |2E
∣∣∣∣∣∂mN(z)
∂Wij
∣∣∣∣∣
2
+ E
∣∣∣∣∂mN(z)
∂Wij
∣∣∣∣2
=σ2
N×(O(
1
M
)+O
(1
M
))
= O(
1
M2
)
which ends the proof. �
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Step 2: Marcenko - Pastur Equation
Proposition
EmN(z)−mN(z) → 0 where mN(z) satisfies:
mN(z) =−1
z[1 + σ2cNmN(z)− σ2(1−cN )
z
] , cN =M
N.
Proof: The mere definition of the resolvent yields
Q = − I
z+
QWW∗
z,
hence
EQr ,i = −δriz
+E(QWW∗)r ,i
z,
where δri stands for the Kronecker symbol.
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Step 2: Marcenko - Pastur Equation (proof I)
Write
E(QWW∗)r ,i =N∑
j=1
M∑
s=1
E(Qr ,sWs,jWij
)
Applying the integration by parts formula yields
E(Qr ,sWs,jWij
)= E |Ws,j |2 E
[∂
∂Ws,j
(Qr ,sWi ,j
)]
=σ2
N
[δsiE(Qr ,s)− E
((Qwj)rQs,sWi ,j
))]
Summing over s and then over j yields:
E (QWW∗)r ,i = σ2EQr ,i − σ2cNE[mN (QWW∗)r ,i
]
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Step 2: Marcenko - Pastur Equation (proof II)
Taking r = i , summing over i and dividing by M yields:
E
(1
MtrQWW∗
)= σ2EmN − σ2cE
[mN
(1
MtrQWW∗
)]
As QWW∗ = I+ zQ, we obtain:
1 + zEmN = σ2EmN − σ2cE [mN (1 + zmN)]
Using Poincare-Nash inequality enables the following decorrelation:
E [mN (1 + zmN)] = (EmN) (1 + zEmN) +O(
1
M2
)
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Step 2: Marcenko - Pastur Equation (proof III)
Gathering the previous results yields:
(1 + zEmN)(1 + σ2cNEmN
)= σ2EmN +O
(1
M2
)
Asymptotically, EmN −mN → 0 which satisfies:
(1 + zmN)(1 + σ2cNmN
)= σ2mN ,
which also writes:
mN(z) =−1
z[1 + σ2cNmN(z)− σ2(1−cN )
z
] .
This ends the proof. �
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Step 3: Marcenko - Pastur Probability distribution
Proposition
The probability distribution associated to the Stieltjes transform mN
admits the density pcN defined as:
pcN (λ) =
{ √(λ−λ−)(λ+−λ)
2πσ2cNif λ ∈ (λ−, λ+)
0 else..
where λ− = σ2(1−√cN)
2 and λ+ = σ2(1 +√cN)
2.
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Step 3: Marcenko - Pastur Probability distribution (proof)
Proof: Solving the equation satisfied by m:
mN(z) = −(z
[1 + σ2cNmN(z)−
σ2(1− cN)
z
])−1
yields
mN(z) =−z + σ2(1− cN) +
√(z − λ−)(z − λ+)
2σ2cNz.
Using the inverse formula yields:
pcN (λ) =1
πlim
y→0+immN(λ+ ιy)
=
{ √(λ−λ−)(λ+−λ)
2πσ2cNif λ ∈ (λ−, λ+)
0 else.,
which is the desired result �
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Concluding remarks
◮ The fact that mN −mN → 0 implies that for f bounded andcontinuous,
1
M
M∑
i=1
f (λi ,N)−∫
f (λ)pcN (λ)dλ→ 0 .
pcN (resp. mN) is a deterministic equivalent of the spectralmeasure LN (resp. mN).
◮ if cN → c∗ ∈ (0,∞), then pcN → pc∗ where pc∗ is obtained byreplacing cN by c∗ and
1
M
M∑
i=1
f (λi ,N) →∫
f (λ)pc∗(λ)dλ .
in this case, the spectral measure converges to pc∗ .
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2 K = 0: An overview of Marcenko and Pastur’s resultsThe Stieltjes transformGaussian toolsMarcenko-Pastur Probability distributionA symmetric view of Marcenko-Pastur equationBehavior of the individual entries of the resolventFiner convergence results
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A companion quantity in MP equation I
Instead of WW∗, consider W∗W. Assuming N ≥ M, both matrices havethe same eigenvalues up to N −M zeroes. The associated Stieltjes
transform therefore writes:
ˆmN(z) =1
Ntr (W∗W − zI)−1
=1
N
(N∑
k=1
1
λk − z− N −M
z
)= cNmN(z)− (1− cN)
1
z
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A companion quantity in MP equation II
As mN −mN → 0, ˆm − mN → 0 which satisfies:
mN(z) = cNmN(z)− (1− cN)1
z.
The inverse Stieltjes transform yields:
(ST )−1(mN) = pcN (λ) and (ST )−1
(−1
z
)= δ0
Hence, we obtain
pcN (dλ) = cNpcN (λ)dλ+ (1− cN)δ0(dλ) ,
where δ0 accounts for the null eigenvalues of W∗W.
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A symmetric view of MP equation
As
mN(z) = −(z
[1 + σ2cNmN(z)−
σ2(1− cN)
z
])−1
mN(z) = cNmN(z)− (1− cN)1
z.
We readily obtain: mN(z) =−1
z(1+σ2mN (z))Similarly, we can obtain the
companion equation: mN(z) =−1
z(1+σ2cNmN(z)). Hence a symmetric
presentation of Marcenko-Pastur equation:
{mN(z) = −1
z(1+σ2mN (z))
mN(z) = −1z(1+σ2cNmN (z))
(1)
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2 K = 0: An overview of Marcenko and Pastur’s resultsThe Stieltjes transformGaussian toolsMarcenko-Pastur Probability distributionA symmetric view of Marcenko-Pastur equationBehavior of the individual entries of the resolventFiner convergence results
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Behavior of the individual entries of the resolvent
Proposition
(diagonal) EQi ,i(z) = mN(z) +O(
1
N3/2
),
(off-diagonal) EQr ,i(z) = O(
1
N3/2
)for r 6= i .
(quadratic form) E u∗Q(z)v = mN(z)(u∗v) +O
(1
N3/2
)
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Behavior of the individual entries of the resolvent (proof I)
Proof: As previously, we have
E (QWW∗)r ,i = σ2EQr ,i − σ2cNE[mN (QWW∗)r ,i
]
Since QWW∗ = I+ zQ, we obtain: (QWW∗)r ,i = δri + zQr ,i Hence:
δri + zEQr ,i = σ2EQr ,i − σ2cNE [mN (δri + zQr ,i)]
= σ2EQr ,i − σ2cNδriEmN − zσ2cNE [mNQr ,i ] .
Poincare-Nash inequality yields
E [mNQr ,i ] = EmNEQr ,i +O(
1
N3/2
)
(follows from the fact that varQr ,i = O(N−1)).
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Behavior of the individual entries of the resolvent (proof II)
◮ If r 6= i then the result is obvious
◮ If r = i , then
1 + zEQi ,i =σ2EQi ,i
1 + σ2cNEmN
+O(
1
N3/2
).
Summing over i and dividing by M yields
1 + zEmN =σ2EmN
1 + σ2cNEmN
+O(
1
N3/2
),
hence the required result.
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2 K = 0: An overview of Marcenko and Pastur’s resultsThe Stieltjes transformGaussian toolsMarcenko-Pastur Probability distributionA symmetric view of Marcenko-Pastur equationBehavior of the individual entries of the resolventFiner convergence results
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Convergence of the extreme eigenvalues
Denote byλ1,N ≥ · · · ≥ λN,N
the ordered eigenvalues of WW∗ and recall that the support ofMarcenko-Pastur distribution is (σ2(1−√
cN)2, σ2(1 +
√cN)
2). Then:
Theorem
If cN → c∗, then the following convergences hold true:
λ1,Na.s.−−−−−→
N,M→∞σ2(1 +
√c∗)
2
λN,Na.s.−−−−−→
N,M→∞σ2(1−√
c∗)2
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Fluctuations of the extreme eigenvalues I
A Central Limit Theorem holds for the largest eigenvalue of matrix WW∗
as N,M → ∞. The limiting distribution is known as Tracy-Widom’sdistribution.
Fluctuations of λ1,N
Let cN → c∗. When correctly centered and rescaled, λ1,N converges to aTracy-Widom distribution:
N2/3
σ2× λ1,N − σ2(1 +
√cN)
2
(1 +√cN)
(1√cN
+ 1)1/3
L−−−−−→N,M→∞
FTW .
The function FTW stands for Tracy-Widom c.d.f. and is preciselydescribed in the following slide.
A similar result holds for λM,N , the smallest eigenvalue of matrix WW∗.
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Fluctuations of the extreme eigenvalues II
Definition of Tracy-Widom’s distribution
The c.d.f. FTW is defined as:
FTW (x) = exp
(−∫ ∞
x
(u − x)q2(u) du
)∀x ∈ R ,
where q solves the Painleve II differential equation:
q”(x) = xq(x) + 2q3(x),
q(x) ∼ Ai(x) as x → ∞.
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1 Problem statement
2 K = 0: An overview of Marcenko and Pastur’s results
3 K fixed: spiked modelsProblem DescriptionMain resultsSome ApplicationsProofs of main results: outline of the approach
4 K may scale with M. Application to the subspace method.
5 Some research prospects
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Signal model
Rcv signal Channel Src signal Noisey1 · · · yN
=
a1 · · · aK
s1
· · ·sK
+
v1 · · · vN
YN = AN SN + VN
M × N M × K K × N M × N
ΣN = N−1/2YN = BN +WN
Recall that noise matrix W has independent CN (0, σ2/N).
We assume here that the number of sources K is ≪ N.
ΣN = Matrix with Gaussian iid elements + fixed rank perturbation.
Asymptotic regime: N → ∞, M/N → c∗, and K is fixed.
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Multiplicative Spiked Model
Assume SN is a random matrix with independent CN (0, 1) elements(Gaussian iid source signals), and AN is deterministic. Then
ΣN =(ANA
∗N + σ2IM
)1/2XN
where XN is M × N with independent CN (0, 1/N) elements.
Consider a spectral factorization
ANA∗N = UN
λ1. . .
λK0
. . .
U∗
N .
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Multiplicative Spiked Model
Let PN be the M ×M matrix
PN = diag
(√λ1 + σ2
σ2, . . . ,
√λK + σ2
σ2, 1, . . . , 1
).
ThenU∗
NΣN = σPNU∗NXN
D= PNWN
where WN is M × N with independent CN (0, σ2/N) elements as above.
PN is a fixed rank perturbation of Identity.⇒ Multiplicative spiked model:
eigenvalues of ΣNΣ∗N ≡ eigenvalues of PNWNW
∗NP
∗N .
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Additive Spiked Model
Assume SN is a deterministic matrix and BN = N−1/2ANSN is suchrank(BN) = K (fixed).We call the model ΣN = BN +WN an additive spiked model.
Impact of BN on spectrum of ΣNΣ∗N in the asymptotic regime ?
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Impact of PN or BN ?
Let FN and FN be the distribution functions associated with the spectralmeasures of ΣNΣ
∗N and WNW
∗N respectively. Then
supx
∣∣∣FN(x) − FN(x)∣∣∣ ≤ 1
Mrank (ΣNΣ
∗N −WNW
∗N) −−−−→
N→∞0
So ΣNΣ∗N and WNW
∗N have the same (Marcenko Pastur) limit spectral
measure, either for the multiplicative or the additive spiked model.
However, ΣNΣ∗N might have isolated eigenvalues.
We shall restrict ourselves to the additive case and study these
isolated eigenvalues as well as the projections on their eigenspaces.
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Spectrum example for ΣNΣ∗N
.
0
2
4
6
8
10
12
0 1 2 3 4 5 6 7 8 9 10.
An eigenvalue histogram for M = 64, N = 3M, and σ2 = 1.ΣN = BN +WN where BN has rank 2 with singular values 2 and 2.5.
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3 K fixed: spiked modelsProblem DescriptionMain resultsSome ApplicationsProofs of main results: outline of the approach
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Notations
Spectral factorizations:
BNB∗N =
u1,N · · · uK ,N
λ1,N
. . .
λK ,N
u1,N · · · uK ,N
∗
where λ1,N ≥ · · · ≥ λK ,N .
Assuming N ≥ M
ΣNΣ∗N =
u1,N · · · uM,N
λ1,N
. . .
λM,N
u1,N · · · uM,N
∗
where λ1,N ≥ · · · ≥ λM,N .
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Main result on the eigenvalues
Theorem 1
Model is ΣN = BN +WN where
BN is a deterministic rank-K matrix such that λk,N → ρk fork = 1, . . . ,K ,
WN is a M × N random matrix with independent CN (0, σ2/N)elements.
Let i ≤ K be the maximum index for which ρi > σ2√c∗. Then for
k = 1, . . . , i ,
λk,Na.s.−−−−→
N→∞γk =
(σ2c∗ + ρk
) (ρk + σ2
)
ρk> σ2(1 +
√c∗)
2
whileλi+1,N
a.s.−−−−→N→∞
σ2(1 +√c∗)
2.
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Main result on the eigenvectors
Theorem 2
Assume the setting of Theorem 1. Assume in addition thatρ1 > ρ1 > · · · > ρi (> σ2
√c∗). For k = 1, . . . , i , let
Πk,N = uk,Nu∗k,N and Πk,N = uk,N u
∗k,N .
Then for any sequence aN of deterministic M × 1 vectors such thatsupN ‖aN‖ <∞,
a∗NΠk,NaN − h(γk)a∗NΠk,NaN
a.s.−−−−→N→∞
0, h(x) =xm(x)2m(x)
(xm(x)m(x))′
and m and m are given by Equations (1) when cN is replaced with c∗.
Generalization to the case of multiple limit eigenvalues ρk is possible.
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3 K fixed: spiked modelsProblem DescriptionMain resultsSome ApplicationsProofs of main results: outline of the approach
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Passive Signal Detection
ΣN = BN +WN , non observable signal + AWGN.
Assume K = 1 source:BN = N−1/2a1,Ns
1N , rank one matrix such that ‖BN‖2 −−−−→
N→∞ρ > 0.
Hypothesis test:
{H0 : ΣN = WN (Noise)H1 : ΣN = BN +WN (Info+Noise)
Generalized Likelihood Ratio Test (GLRT)
TN =λ1,N
M−1 tr (ΣNΣ∗N)
Asymptotic behavior ?
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Passive Signal Detection and Additive Spiked Models
Under either H0 or H1, M−1 tr (ΣNΣ∗N)
a.s.−−−−→N→∞
σ2.
Under H1 (consequence of main result on eigenvalues):◮ If ρ > σ2√c∗, then
λ1,Na.s.−−−−→
N→∞
γ =
(σ2c∗ + ρ
) (ρ+ σ2
)
ρ> σ2(1 +
√c∗)
2,
λ2,Na.s.−−−−→
N→∞
σ2(1 +√c∗)
2.
◮ If ρ ≤ σ2√c∗, then
λ1,Na.s.−−−−→
N→∞
σ2(1 +√c∗)
2.
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Passive Signal Detection and Additive Spiked ModelsWe therefore have
Under H0,TN
a.s.−−−−→N→∞
(1 +√c∗)
2.
Under H1,◮ If ρ > σ2√c∗, then
TNa.s.−−−−→
N→∞
(σ2c∗ + ρ
) (ρ+ σ2
)
σ2ρ> (1 +
√c∗)
2
◮ If ρ ≤ σ2√c∗, then
TNa.s.−−−−→
N→∞
(1 +√c∗)
2.
ρ > σ2√c∗ provides the limit of detectability by the GLRT.
False Alarm Probability can be evaluated with the help of theTracy-Widom law.
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Source localization
Problem
K radio sources send their signals to a uniform array of M antennas duringN signal snapshots.
Estimate arrival angles ϕ1, . . . , ϕK
.
ϕ2
ϕ1
.
Example with two sources
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Source localization with a subspace method (MUSIC)
Model: ΣN = N−1/2ANSN︸ ︷︷ ︸BN
+WN with
AN =[aN(ϕ1) · · · aN(ϕK )
]with aN(ϕ) =
1√M
1eıπ sinϕ
...
eı(M−1)π sinϕ
SN is deterministic, rank(SN) = K .
Let ΠN be the orthogonal projection matrix on the span of AA∗, orequivalently, on the eigenspace of EΣΣ∗ = BB∗ + σ2IM associated withthe eigenvalues > σ2 (“signal subspace”). Let Π⊥
N = IM −ΠN be theorthogonal projector on the “noise subspace”.
MUSIC algorithm principle
aN(ϕ)∗Π⊥
NaN(ϕ) = 0 ⇔ ϕ ∈ {ϕ1, . . . , ϕK}.
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MUSIC algorithm
Traditional MUSIC: angles are estimated as local minima of
aN(ϕ)∗Π
⊥NaN(ϕ)
where ΠN is the orthogonal projection matrix on the eigenspace associated
with the K largest eigenvalues of ΣΣ∗ and Π⊥N = IM − ΠN .
Asymptotic behavior of aN(ϕ)∗Π
⊥NaN(ϕ) well known when M is fixed and
N → ∞.
Behavior in our asymptotic regime ?
Is it possible to improve the traditional estimator and to adapt it toour asymptotic regime ?
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MUSIC algorithm and the spiked additive model
Modified MUSIC estimator: application of Theorem 2
Assume that lim infN
λK ,N > σ2√c∗. Then
aN(ϕ)∗ΠNaN(ϕ)−
K∑
k=1
|aN(ϕ)∗uk,N |2
h(λk,N)
a.s.−−−−→N→∞
0
uniformly on ϕ ∈ [0, π].
Modification of the traditional estimator
a(ϕ)∗Π⊥a(ϕ) = a(ϕ)∗(
M∑
k=1
uk u∗k −Π
)a(ϕ)
N large≃ a(ϕ)∗(
K∑
k=1
(1− 1
h(λk)
)uk u
∗k +
M∑
k=K+1
uk u∗k
)a(ϕ)
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3 K fixed: spiked modelsProblem DescriptionMain resultsSome ApplicationsProofs of main results: outline of the approach
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Eigenvalues: principle of the proof of Theorem 1
We follow the approach of Benaych-Georges and Nadakuditi’2011.We study the isolated eigenvalues of ΣΣ∗, or equivalently, the isolatedsingular values of Σ.
A matrix algebraic lemma
Let A be a M × N matrix. Then σ1, . . . , σM∧N are the singular values ofA if and only if
σ1, . . . , σM∧N ,−σ1, . . . ,−σM∧N , 0, . . . , 0︸ ︷︷ ︸|N −M|
are the eigenvalues of
A =
[0 A
A∗ 0
]
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Eigenvalues: principle of the proof of Theorem 1Drop index N. Let B = U
√ΛV∗, Λ = diag (λ1, . . . , λK ) be a spectral
factorisation of B. Write
Σ =
[0 Σ
Σ∗ 0
]=
[0 W
W∗ 0
]+
[U 0
0 V√Λ
] [0 IKIK 0
] [U∗ 0
0√ΛV∗
]
= W + CJC∗.
Assume λ 6∈ spectrum(WW∗) and λ ∈ spectrum(ΣΣ∗) or equivalently
det(W −
√λIM+N
)6= 0 and det
(Σ−
√λIM+N
)= 0.
We have
det (Σ− xI) = det (W − xI+ CJC∗)
= det (W − xI) det(I2K + JC∗ (W − xI)−1
C)
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Eigenvalues: principle of the proof of Theorem 1
Using inversion formula for partitioned matrices,
(W − xI)−1 =
[−xI W
W∗ −xI
]−1
=
[xQ(x2) WQ(x2)
Q(x2)W∗ xQ(x2)
]
where we recall that Q(x) = (WW∗ − xI)−1, and where we setQ(x) = (W∗W − xI)−1.
Hence√λ is a zero of
det(I2K + JC∗ (W − xI)−1
C)
= (−1)K det
[xU∗Q(x2)U IK +U∗WQ(x2)V
√Λ
IK +√ΛV∗Q(x2)W∗U x
√ΛV∗Q(x2)V
√Λ
]
︸ ︷︷ ︸H(x)
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Eigenvalues: principle of the proof of Theorem 1
When x2 > σ2(1 +√c∗)2, Q(x2) and Q(x2) are well defined for N large,
because ‖WW∗‖ a.s.−−−−→N→∞
σ2(1 +√c∗)2.
By the approach developed in the previous chapter
U∗Q(x2)Ua.s.−−−−→
N→∞m(x2)IK , V∗Q(x2)V
a.s.−−−−→N→∞
m(x2)IK , and
V∗Q(x2)W∗Ua.s.−−−−→
N→∞0,
hence
H(x)a.s.−−−→
n→∞H(x) =
[xm(x2)IK IK
IK xm(x2)Γ
]where Γ =
ρ1
. . .
ρK
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Eigenvalues: principle of the proof of Theorem 1
Consider the equation
detH(√x) =
K∏
k=1
(xm(x)m(x)ρk − 1) = 0 . (2)
Let γ+ = σ2(1 +√c∗)2. From the general properties of the Stieltjes
Transforms, function G (x) = xm(x)m(x) decreases from G (γ++) tozero for x ∈ (γ+,∞).
Recall the ρk ’s are arranged in decreasing order. Assumeρk > 1/G (γ++). Then the k th largest zero γk of (2) (which satisfiesG (γk) = 1/ρk) will satisfy γk > γ+.
In that situation, due to det H →as detH outside the eigenvalue bulk,we infer that λk →as γk . Otherwise, λk →as γ+.
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Illustration
.
G(x)
G(γ++)
γ+ γk
1/ρk
.
Exploiting the expressions of m(z) and m(z) (Stieltjes Transforms of M-Pdistributions), condition ρk > 1/G (γ++) can be rewritten ρk > σ2
√c∗.
In this case, solving G (γk) = 1/ρk gives γk =(σ2c∗ + ρk
) (ρk + σ2
)/ρk .
Hence Theorem 1.
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Eigenvectors: principle of the proof of Theorem 2
Matrix algebraic lemma (cont’d)
A pair (u, v) of unit norm vectors is a pair of (left,right) singular vectors ofthe M × N matrix A associated with the singular value σ if and only if
2−1/2
[u
v
]is a unit norm eigenvector of
A =
[0 A
A∗ 0
]
associated with the eigenvalue σ.
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Eigenvectors: principle of the proof of Theorem 2
Quadratic form a∗Πka can be written as a Cauchy-integral: using theprevious lemma,
a∗Πka =−1
ıπ
∮
Ck
[a∗ 0
]([ 0 Σ
Σ∗ 0
]− zIM+N
)−1 [a
0
]dz
where path Ck encloses eigenvalue√λk .
Recalling that
[0 Σ
Σ∗ 0
]=
[0 W
W∗ 0
]+ CJC∗, we obtain using the
inversion formula for partitioned matrices
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Eigenvectors: principle of the proof of Theorem 2
a∗Πka =−1
ıπ
∮
Ck
[a∗ 0
] ([ 0 W
W∗ 0
]− zI
)−1 [a
0
]dz
︸ ︷︷ ︸= 0 for large N
+1
ıπ
∮
Ckb∗(z)H(z)−1b(z) dz
where
b(z) =
[zU∗Q(z2)√ΛV∗Q(z2)W∗
]a ,
and recall that H(z) =
[zU∗Q(z2)U IK +U∗WQ(z2)V
√Λ
IK +√ΛV∗Q(z2)W∗U z
√ΛV∗Q(z2)V
√Λ
].
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Eigenvectors: principle of the proof of Theorem 2
Let
b(z) =
[zm(z2)U∗a
0
]and recall H(z) =
[zm(z2)IK IK
IK zm(z2)Γ
]
Since λk →a.s. γk =(σ2c∗ + ρk
) (ρk + σ2
)/ρk , we replace Ck with a
deterministic path Ck centered around γk , and
a∗ΠkalargeN≃ 1
ıπ
∮
Ck
b∗(z)H(z)−1b(z) dz
=γkm(γk)
2m(γk)
(γkm(γk)m(γk))′a∗Πka
using the residue theorem.
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1 Problem statement
2 K = 0: An overview of Marcenko and Pastur’s results
3 K fixed: spiked models
4 K may scale with M. Application to the subspace method.Motivation.The ”asymptotic” limit eigenvalue distribution µNContours enclosing only the eigenvalue 0 of BNB
HN
The G-MUSIC algorithm.
5 Some research prospects
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4 K may scale with M. Application to the subspace method.Motivation.The ”asymptotic” limit eigenvalue distribution µNContours enclosing only the eigenvalue 0 of BNB
HN
The G-MUSIC algorithm.
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YN = ASN + VN
A M × K deterministic, the source K × N matrix SN deterministic.
K and M are possibly of the same order of magnitude: K may scalewith N in contrast with the context of spiked models.
After normalization by√N :
ΣN = BN +WN
BN = ASN√N
deterministic, Rank(BN) = K = K (N) < M = M(N)
WN complex Gaussian i.i.d. matrix, E|Wi ,j |2 = σ2
N
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ΣN = BN +WN
Noise subspace: Orthogonal of the range of BN = orthogonal of therange of A under mild conditions,
Orthogonal projection matrix Π⊥N
Estimate consistently aHΠ⊥Na for each unit norm M–dimensional
deterministic vector a
The conventional estimate aHΠ⊥Na is not consistent:
aHΠ⊥Na− aHΠ⊥
Na does not converge to 0 if M and N converge to ∞at the same rate
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4 K may scale with M. Application to the subspace method.Motivation.The ”asymptotic” limit eigenvalue distribution µNContours enclosing only the eigenvalue 0 of BNB
HN
The G-MUSIC algorithm.
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Characterization of the limit eigenvalue distribution µN
Dozier-Silverstein 2007: It exists a deterministic probability measureµN carried by R+ such that
1M
∑Mk=1 δ(λ − λk,N)− µN → 0 weakly almost surely
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Characterization of the limit eigenvalue distribution µN
Dozier-Silverstein 2007: It exists a deterministic probability measureµN carried by R+ such that
1M
∑Mk=1 δ(λ − λk,N)− µN → 0 weakly almost surely
How to characterize µN
Stieltjes transform mN(z) =∫R+
µN (dλ)λ−z
defined on C− R+
mN(z) :=1MTrTN(z) with
TN(z) =(
BNB∗
N
1+σ2cNmN(z)− z(1 + σ2cNmN(z))IM + σ2(1− cN)IM
)−1.
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Equivalent form of the equation
mN(z) is solution of the equation
mN(z)
1 + σ2cNmN(z)=
1
MTrace(BNB
∗N − wN(z)IM)−1 = fN(wN(z))
wN(z) = z(1 + σ2cNmN(z))2 − σ2(1− cN)(1 + σ2cNmN(z))
fN(w) = 1MTrace(BNB
∗N − w IM)−1 = 1
M
∑Mk=1
1λk,N−w
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Equivalent form of the equation
mN(z) is solution of the equation
mN(z)
1 + σ2cNmN(z)=
1
MTrace(BNB
∗N − wN(z)IM)−1 = fN(wN(z))
wN(z) = z(1 + σ2cNmN(z))2 − σ2(1− cN)(1 + σ2cNmN(z))
fN(w) = 1MTrace(BNB
∗N − w IM)−1 = 1
M
∑Mk=1
1λk,N−w
Convergence results: QN(z) = (ΣNΣ∗N − zIM)−1
1MTrQN(z) = mN(z) ≍ mN(z) =
1MTrTN(z)
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Equivalent form of the equation
mN(z) is solution of the equation
mN(z)
1 + σ2cNmN(z)=
1
MTrace(BNB
∗N − wN(z)IM)−1 = fN(wN(z))
wN(z) = z(1 + σ2cNmN(z))2 − σ2(1− cN)(1 + σ2cNmN(z))
fN(w) = 1MTrace(BNB
∗N − w IM)−1 = 1
M
∑Mk=1
1λk,N−w
Convergence results: QN(z) = (ΣNΣ∗N − zIM)−1
1MTrQN(z) = mN(z) ≍ mN(z) =
1MTrTN(z)
Hachem et al.(2010), for ‖dN‖ = 1,
d∗NQN(z)dN ≍ d∗NTN(z)dN .
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Properties of µN , cN = MN< 1
Dozier-Silverstein-2007
For each x ∈ R, limz→x ,z∈C+ mN(z) = mN(x) exists
x → mN(x) continuous on R, continuously differentiable on R\∂SN
µN(dλ) absolutely continuous, density 1π Im(mN(x))
SN support of µN . Int(SN) = {x ∈ R, Im(mN(x)) > 0}
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Characterization of the support SN of µN .
Reformulation of Dozier-Silverstein 2007 inVallet-Loubaton-Mestre-2010
Function φN(w) defined on R byφN(w) = w(1− σ2cN fN(w))2 + σ2(1− cN)(1− σ2cN fN(w))
φN has 2Q positive extrema with preimages
w(N)1,− < w
(N)1,+ < w
(N)2,− < . . .w
(N)Q,− < w
(N)Q,+. These extrema verify
x(N)1,− < x
(N)1,+ < x
(N)2,− < . . . x
(N)Q,− < x
(N)Q,+.
SN = [x(N)1,− , x
(N)1,+ ] ∪ . . . [x(N)
Q,−, x(N)Q,+]
Each eigenvalue λl ,N of BNB∗N belongs to an interval (w
(N)k,− ,w
(N)k,+)
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w
φN (w)
w−
1,N
Support SN
x+
3,N
x−
3,N
x+
2,N
x−
2,N
x+
1,N
x−
1,N
w+
1,N w−
2,N
λ3,N λ2,N λ1,N
w+
2,Nw−
3,Nw+
3,N
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If cN is small enough or σ2 small enough, there are Q = K + 1clusters nearly centered around σ2 and (λk + σ2)k=1,...,K .
If cN or σ2 increases, certain clusters merge, and Q < K + 1.
An eigenvalue λk,N of BNB∗N is said to be associated to the cluster
[x−q,N , x+q,N ] if λk,N ∈]w−
q,N ,w+q,N [.
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Illustration (I).
The parameters.
σ2 = 2
Eigenvalues of BNB∗N : 0 and 5 with multiplicity M
2
Eigenvalues of BNB∗N + σ2I : 2 and 7 with multiplicity M
2
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Illustration (I).
The parameters.
σ2 = 2
Eigenvalues of BNB∗N : 0 and 5 with multiplicity M
2
Eigenvalues of BNB∗N + σ2I : 2 and 7 with multiplicity M
2
Remark
fN(w) = 12
(− 1
w+ 1
5−w
)independent of M,N
µN does not depend on M,N if cN = MN
= c independent of M,N
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Illustration (II).
c = MN
= 0.05
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Illustration (III).
c = MN
= 0.2
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Illustration (IV).
c = MN
= 0.5
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SN in the context of spiked models.
Assumptions: cN → c∗, λk,N → ρk > σ2√c∗ for k = 1, . . . ,K ,
ρk 6= ρl .
λk,N → γk =(σ2c∗+ρk)(ρk+σ2)
ρkfor k = 1, . . . ,K
Q = K + 1 clusters
[x−1,N , x+1,N ] = [σ2(1−√
cN)2 −O( 1
N), σ2(1 +
√cN)
2 −O( 1N)]
[x−k,N , x+k,N ] = [ψ(λK+2−k,N , cN)−O( 1√
N), ψ(λK+2−k,N , cN)+O( 1√
N)]
for k = 2, . . . ,K + 1
ψ(λ, c) =(σ2c+λ)(λ+σ2)
λ so that ψ(λK+2−k,N , cN) close fromψ(ρK+2−k , c∗) = γK+2−k .
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Illustration
c = MN
= 0.5,N = 100, K = 2, σ2 = 1
−4 −2 0 2 4 6 8 10 12 14 160
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
DensityEigenvalues
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4 K may scale with M. Application to the subspace method.Motivation.The ”asymptotic” limit eigenvalue distribution µNContours enclosing only the eigenvalue 0 of BNB
HN
The G-MUSIC algorithm.
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Some useful properties of wN(z)
wN(z) = z(1 + σ2cNmN(z))2 − σ2(1− cN)(1 + σ2cNmN(z)).
Im(wN(z)) > 0 if Im(z) > 0
Int(SN) = {x , Im(wN(x)) > 0}wN(x) is real and increasing on each component of Sc
N
wN(x−q,N) = w−
q,N ,wN(x+q,N) = w+
q,N
wN(x) is continuous on R and continuously differentiable on R\∂SN
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Illustration of the behaviour of x → wN(x)
Illustration 2 clusters.
Re{w(x)}
Im{w(x)}
w(x−
1) = w
−
1w(x
+
1) = w
+
1
w(x−
2) = w
−
2w(x
+
2)= w
+
2
λ3λ4
0λ2 λ1
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In the case of the MP distribution, BN = 0
wN(x) is real and increasing on (−∞, σ2(1−√cN)
2)
wN(σ2(1−√
cN)2) = −σ2√cN
|wN(x)| = σ2√cN if x ∈ [σ2(1−√
cN)2, σ2(1 +
√cN)
2
wN(1 +√cN)
2) = σ2√cN
wN(x) is real and increasing on (σ2(1 +√cN)
2,+∞)
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Illustration in the spiked case K = 2, N = 100, M = 50, σ2 = 1
−2 0 2 4 6 8 10−1
0
1
2
3
4
5Im
(wN
(x))
Re(wN
(x))Hachem, Loubaton and Najim (EUSIPCO) Large random matrices August / September 2011 113 / 133
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4 K may scale with M. Application to the subspace method.Motivation.The ”asymptotic” limit eigenvalue distribution µNContours enclosing only the eigenvalue 0 of BNB
HN
The G-MUSIC algorithm.
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Valid under the following hypotheses.
Assumptions.
0 is the unique eigenvalue associated with [x−1,N , x+1,N ] for each N large
enough,
0 < lim infN x−1,N < lim supN x+1,N < lim infN x−2,N
t−1
x−
1,N t+1
x+
1,N t−2
x−
2,Nx+
Q ,N...
for all N large enough , t−1 , t+1 , t
−2 independent of N
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Consequences of the assumptions
almost surely for N large enough
λK+1,N , . . . , λM,N ∈ (t−1 , t+1 ) and λ1,N , . . . , λK ,N > t−2 ,
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Consequences of the assumptions
almost surely for N large enough
λK+1,N , . . . , λM,N ∈ (t−1 , t+1 ) and λ1,N , . . . , λK ,N > t−2 ,
almost surely for N large enough,
ωK+1,N , . . . , ωM,N ∈ (t−1 , t+1 ) and ω1,N , . . . , ωK ,N > t−2
with ω1,N ≥ . . . ≥ ωM,N the solutions of the equation1 + σ2cNmN(z) = 0 with mN(z) =
1MTrQN(z)
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Consequences of the assumptions
For y > 0, we define the domain
Ry ={u + iv : u ∈ [t−1 − δ, t+1 + δ], v ∈ [−y , y ]
}.
Then, if t+1 + δ < t−2 , Cy = wN(∂Ry ) encloses 0 and no othereigenvalue of BNB
∗N for N large enough.
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Consistent estimation of ηN = aNΠ⊥NaN .
From residues theorem:
ηN =1
2πı
∮
C−
y
a∗N (BNB∗N − λIM)−1
aNdλ,
ηN =1
2πı
∮
∂R−
y
a∗N (BNB∗N − wN(z)IM)−1
aNw′N(z)dz
ηN =1
2πı
∮
∂R−
y
a∗NTN(z)aNw ′N(z)
1 + σ2cNmN(z)dz
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The integrand can be estimated consistently.
gN(z) = a∗NTN(z)aNw ′
N(z)
1+σ2cNmN (z)
From the previous result, we have the following convergence onC− SN
mN(z) ≍ mN(z) =1
MTrQN(z) and a∗NTN(z)aN ≍ a∗NQN(z)aN
with QN(z) = (ΣNΣ∗N − zIM)−1.
Let gN(z) := a∗NQN(z)aNw ′
N(z)
1+σ2cN mN(z)with
wN(z) = z(1 + σ2cNmN(z))2 − σ2cN(1 + σ2cNmN(z)). gN(z) has no
pole on ∂Ry and
∣∣∣∣∣1
2πı
∮
∂R−
y
(gN(z)− gN(z))dz
∣∣∣∣∣→ 0 a.s.,
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The new consistent estimator.
ηN,new = 12πı
∮∂R−
ya∗NQN(z)aN
w ′
N(z)
1+σ2cN mN (z)dz
Integral can be solved using the residue’s theorem
ηN,new = a∗N
(∑Mk=1 ξk,N uk,N u
∗k,N
)aN with (ξk,N) depending on
λ1,N , . . . , λM,N and ω1,N , . . . , ωM,N .
ηN,new depend on the (uk,N u∗k,N)k=K+1,...,M and on the
(uk,N u∗k,N)k=1,...,K
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Numerical evaluations.
Comparisons between:
The traditional subspace method
The spike subspace method
The improved subspace method
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Experiment 1
Parameters
a(ϕ) = 1√M[1, expıπ sin(ϕ), . . . , expı(M−1)π sin(ϕ)]T
source signals are AR(1) processes with correlation coefficient of 0.9
K = 2,M = 20,N = 40, ϕ1 = 16, ϕ2 = 18
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Mean of the MSE of ϕ1 and ϕ2 versus SNR.
0 5 10 15 20 25 3010
−3
10−2
10−1
100
101
102
SNR
MS
E
Trad−MUSICG−MUSICSpike−MUSICCRB
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Mean of the MSE of the a(ϕi)HΠ
⊥Na(ϕi) versus SNR
0 5 10 15 20 25 3010
−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
SNR
MS
E
Trad−MUSICG−MUSICSpike−MUSIC
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Experiment 2
Parameters
K = 5,M = 20,N = 40
angles equal to −20,−10, 0, 10, 20
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Mean of the MSE of the a(ϕi)HΠ
⊥Na(ϕi) versus SNR
0 5 10 15 20 25 3010
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
101
SNR
MS
E
Trad−MUSICG−MUSICSpike−MUSIC
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1 Problem statement
2 K = 0: An overview of Marcenko and Pastur’s results
3 K fixed: spiked models
4 K may scale with M. Application to the subspace method.
5 Some research prospects
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Future applications
G-estimation of other parameters: number of sources, powerdistribution, ...Applications: cognitive radio or passive network metrology.
Application of the spiked models for local failure detection/diagnosisin large data or power networks.
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Methodological future research
Spiked models:◮ Performance of tests for isolated eigenvalues, e.g. with the help of
large deviations theory.◮ Design and evaluation of sphericity tests.
G-estimation:◮ Extension of the G-estimation techniques to other matrix models.◮ Consistency and fluctuations of estimates.
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Bibliography
F. Benaych-Georges and R. R. Nadakuditi, ”The singular values andvectors of low rank perturbations of large rectangular randommatrices”. ArXiv e-prints, 2011.
P. Bianchi, M. Debbah, M. Maida, J. Najim, ”Performance ofStatistical Tests for Source Detection using Random Matrix Theory”,IEEE Trans. IT, 57 (4), 2011.
R. Couillet, J. W. Silverstein, Z. Bai, M. Debbah, ”Eigen-Inference forEnergy Estimation of Multiple Sources”, IEEE Trans. IT., 57(4),2011.
M. Debbah, W. Hachem, Ph. Loubaton and M. de Courville, ”MMSEAnalysis of Certain Large Isometric Random Precoded Systems”,IEEE Trans. IT, 49 (5), 2003.
R.B. Dozier and J.W. Silverstein, ”On the empirical distribution ofeigenvalues of large dimensional information plus noise typematrices”, JMVA, 98 (4), 2007.
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Bibliography
R.B. Dozier and J.W. Silverstein, ”Analysis of the limiting spectraldistribution of large dimensional information plus noise typematrices”, JMVA, 98 (6), 2007.
J. Dumont, W. Hachem, S. Lasaulce, Ph. Loubaton and J. Najim,”On the Capacity Achieving Covariance Matrix for Rician MIMOChannels: An Asymptotic Approach”, IEEE Trans. IT, 56 (3), 2010.
W. Hachem, Ph. Loubaton, J. Najim and P. Vallet, ”On bilinearforms based on the resolvent of large random matrices”, ArXive-prints, 2010.
N. El Karoui, ”Spectrum estimation for large dimensional covariancematrices using random matrix theory”, Ann. Statist., 36 (6), 2008.
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Bibliography
X. Mestre, ”Improved Estimation of Eigenvalues and Eigenvectors ofCovariance Matrices Using Their Sample Estimates”, IEEE Trans. IT,54 (11), 2008.
X. Mestre and M.A. Lagunas, ”Modified subspace algorithms for DoAestimation with large arrays”, IEEE Trans. SP, 56 (2), 2008.
B. Nadler, ”Nonparametric Detection of Signals by InformationTheoretic Criteria: Performance Analysis and an ImprovedEstimator”, IEEE Trans. SP, 58 (5), 2010.
L.A. Pastur, ”A Simple Approach to the Global Regime of GaussianEnsembles of Random Matrices”, Ukrainian Math. J., 57 (6), 2005.
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