Interference Alignment via Message-Passing
Interference
Alignment via
Message-Passing
M. Guillaud
Motivation
Maxime GUILLAUD
Communication Problem
Interference Alignment
Message-Passing
GDL
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
Huawei Technologies
Mathematical and Algorithmic Sciences Laboratory, Paris
Numerical Results
Conclusion
[email protected]
http://research.mguillaud.net/
Optimisation Géométrique sur les Variétés
Réunion GdR ISIS, Paris, 21 Nov. 2014
1/21
Outline
Interference
Alignment via
Message-Passing
M. Guillaud
Motivation
Communication Problem
Interference Alignment
Message-Passing
I
Motivation of the Interference Alignment Problem
I
Message-passing and optimization
GDL
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
Numerical Results
I
Numerical Results
I
Conclusion
Conclusion
2/21
Interference
Alignment via
Message-Passing
Communication Problem Motivation
M. Guillaud
Motivation
Communication Problem
Interference Alignment
Message-Passing
I
K transmitters (Tx), K receivers (Rx)
GDL
Tx 1
I
Each equipped with an antenna array;
each antenna sends/receives a
complex scalar
I
Rx i is only interested in the message
from the corresponding Tx i
I
Other transmitters create (unwanted)
interference
Rx 1
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
Tx 2
Rx 2
Tx 3
Rx 3
Tx K
Rx K
Numerical Results
Conclusion
3/21
Interference
Alignment via
Message-Passing
Communication Problem Motivation
M. Guillaud
I
I
At each (discrete) time instant, Tx j
sends a (N-dimensional) signal x j , Rx
i receives y i (M-dimensional)
The gains of the wireless propagation
channel between Tx j and Rx i is Hij
(M × N matrix)
Motivation
Tx 1
Rx 1
Communication Problem
Interference Alignment
Message-Passing
Tx 2
Rx 2
Tx 3
Rx 3
GDL
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
Numerical Results
Conclusion
Tx K
yi =
K
X
Hij x j
Rx K
∀i = 1 . . . K
j=1
I
Rx i wants to infer x i from y i (all Hij are assumed known at
all nodes)
4/21
Interference
Alignment via
Message-Passing
Communication Problem Motivation
M. Guillaud
I
I
At each (discrete) time instant, Tx j
sends a (N-dimensional) signal x j , Rx
i receives y i (M-dimensional)
The gains of the wireless propagation
channel between Tx j and Rx i is Hij
(M × N matrix)
Motivation
Tx 1
Rx 1
Communication Problem
Interference Alignment
Message-Passing
Tx 2
Rx 2
Tx 3
Rx 3
GDL
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
Numerical Results
Conclusion
Tx K
yi =
K
X
Hij x j
Rx K
∀i = 1 . . . K
j=1
I
Rx i wants to infer x i from y i (all Hij are assumed known at
all nodes)
4/21
Interference
Alignment via
Message-Passing
Communication Problem Motivation
M. Guillaud
I
I
At each (discrete) time instant, Tx j
sends a (N-dimensional) signal x j , Rx
i receives y i (M-dimensional)
The gains of the wireless propagation
channel between Tx j and Rx i is Hij
(M × N matrix)
Motivation
Tx 1
Rx 1
Communication Problem
Interference Alignment
Message-Passing
Tx 2
Rx 2
Tx 3
Rx 3
GDL
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
Numerical Results
Conclusion
Tx K
yi =
K
X
Hij x j
Rx K
∀i = 1 . . . K
j=1
I
Rx i wants to infer x i from y i (all Hij are assumed known at
all nodes)
4/21
Interference
Alignment via
Message-Passing
Inteference Alignment
Simple, linear transmission scheme that mitigates interference:
I
Tx signals are restricted to a d-dimensional subspace of the
N-dimensional space of the antenna array, spanned by a
truncated unitary matrix Vj . s j is the data to transmit:
M. Guillaud
Motivation
Communication Problem
Interference Alignment
Message-Passing
GDL
(Vj ∈ GN,d )
x j = Vj s j
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
Numerical Results
I
At Rx i, signal is projected onto a d-dimensional subspace
spanned by a truncated unitary matrix Ui (∈ GM,d )
s 0 i = U†i y i =
K
X
Conclusion
U†i Hij Vj s j
j=1
I
Choose the Ui , Vj such that U†i Hij Vj vanishes for i 6= j
(interference) but not for i = j
5/21
Interference
Alignment via
Message-Passing
Inteference Alignment
Simple, linear transmission scheme that mitigates interference:
I
Tx signals are restricted to a d-dimensional subspace of the
N-dimensional space of the antenna array, spanned by a
truncated unitary matrix Vj . s j is the data to transmit:
M. Guillaud
Motivation
Communication Problem
Interference Alignment
Message-Passing
GDL
(Vj ∈ GN,d )
x j = Vj s j
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
Numerical Results
I
At Rx i, signal is projected onto a d-dimensional subspace
spanned by a truncated unitary matrix Ui (∈ GM,d )
s 0 i = U†i y i =
K
X
Conclusion
U†i Hij Vj s j
j=1
I
Choose the Ui , Vj such that U†i Hij Vj vanishes for i 6= j
(interference) but not for i = j
5/21
Interference
Alignment via
Message-Passing
Inteference Alignment
Simple, linear transmission scheme that mitigates interference:
I
Tx signals are restricted to a d-dimensional subspace of the
N-dimensional space of the antenna array, spanned by a
truncated unitary matrix Vj . s j is the data to transmit:
M. Guillaud
Motivation
Communication Problem
Interference Alignment
Message-Passing
GDL
(Vj ∈ GN,d )
x j = Vj s j
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
Numerical Results
I
At Rx i, signal is projected onto a d-dimensional subspace
spanned by a truncated unitary matrix Ui (∈ GM,d )
s 0 i = U†i y i =
K
X
Conclusion
U†i Hij Vj s j
j=1
I
Choose the Ui , Vj such that U†i Hij Vj vanishes for i 6= j
(interference) but not for i = j
5/21
Mathematical Formulation
Interference
Alignment via
Message-Passing
M. Guillaud
Motivation
Communication Problem
I
Given the M × N matrices {Hij }i,j∈{1,...K } and d < M, N
I
find {Ui }i∈{1,...K } in GM,d and {Vi }i∈{1,...K } in in GN,d
I
such that U†i Hij Vj = 0 ∀i 6= j.
Interference Alignment
Message-Passing
GDL
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
Numerical Results
I
Depending on the relative values of M, N, K and d, the
problem can be trivial, difficult, or provably impossible to
solve.
I
Example of non-trivial case: d = 2, M = N = 4, K = 3.
Conclusion
6/21
Mathematical Formulation
Interference
Alignment via
Message-Passing
M. Guillaud
Motivation
Communication Problem
I
Given the M × N matrices {Hij }i,j∈{1,...K } and d < M, N
I
find {Ui }i∈{1,...K } in GM,d and {Vi }i∈{1,...K } in in GN,d
I
such that U†i Hij Vj = 0 ∀i 6= j.
Interference Alignment
Message-Passing
GDL
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
Numerical Results
I
Depending on the relative values of M, N, K and d, the
problem can be trivial, difficult, or provably impossible to
solve.
I
Example of non-trivial case: d = 2, M = N = 4, K = 3.
Conclusion
6/21
Interference
Alignment via
Message-Passing
Matrix Decomposition Formulation
M. Guillaud
Motivation
Communication Problem
Interference Alignment
Message-Passing
GDL
Min-Sum

U†1


..
|
{z
.

U†K
Kd×KM
}|
H11
..
.
HK 1
...
..
.
...
{z
KM×KN
H1K
..
.
HKK


V1

..
}|
{z

∗
..
=
.
0
VK
KN×Kd
}
0

IA via Min-Sum
Implementation challenges
Distributedness
.

∗
Numerical Results
Conclusion
7/21
Interference
Alignment via
Message-Passing
Available Solutions
M. Guillaud
Motivation
Communication Problem
State of the art:
I
I
Interference Alignment
No closed-form solution known for general dimensions
1
Iterative solutions exist but have some undesirable properties
Message-Passing
GDL
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
We propose to use a message-passing (MP) algorithm:
I
distributed solution
I
use local data (Hij is known at Rx i and Tx j)
Numerical Results
Conclusion
1
K. Gomadam, V.R. Cadambe, S.A. Jafar, A Distributed Numerical
Approach to Interference Alignment and Applications to Wireless Interference
Networks, IEEE Trans. Inf. Theory, Jun 2011
8/21
Interference
Alignment via
Message-Passing
Available Solutions
M. Guillaud
Motivation
Communication Problem
State of the art:
I
I
Interference Alignment
No closed-form solution known for general dimensions
1
Iterative solutions exist but have some undesirable properties
Message-Passing
GDL
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
We propose to use a message-passing (MP) algorithm:
I
distributed solution
I
use local data (Hij is known at Rx i and Tx j)
Numerical Results
Conclusion
1
K. Gomadam, V.R. Cadambe, S.A. Jafar, A Distributed Numerical
Approach to Interference Alignment and Applications to Wireless Interference
Networks, IEEE Trans. Inf. Theory, Jun 2011
8/21
Generalized Distributive Law
Interference
Alignment via
Message-Passing
M. Guillaud
I
Efficiently compute functions involving many variables that
can be decomposed (factorized) into terms involving subsets
of the variables
I
Variable–Factor dependencies captured by a bipartite graph
I
General MP formulation for an arbitrary commutative
semi-ring2
Motivation
Communication Problem
Interference Alignment
Message-Passing
GDL
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
Numerical Results
Conclusion
2
Aji and McEliece, The Generalized Distributive Law, IEEE Trans. Inf.
Theory, vol. 46, no. 2, Mar. 2000.
9/21
Generalized Distributive Law
Interference
Alignment via
Message-Passing
M. Guillaud
Motivation
Communication Problem
Interference Alignment
Message-Passing
I
Algorithm operates by “propagating” messages on the graph
I
If the graph has no cycle, computation is exact and
terminates in finite number of steps
GDL
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
Numerical Results
I
Otherwise, iterate and hope that it converges to something
reasonable
I
BCJR decoder, Viterbi, FFT algorithms as special cases
Conclusion
10/21
Min-Sum Algorithm3
Interference
Alignment via
Message-Passing
M. Guillaud
I
GDL applied to (R, min, +)
I
Example:
Motivation
Communication Problem
Interference Alignment
C (x1 , x2 , x3 , x4 ) = Ca (x1 , x2 , x3 ) + Cb (x2 , x4 ) + Cc (x3 , x4 )
Message-Passing
GDL
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
Numerical Results
Conclusion
I
Min-Sum can solve efficiently
min
x1 ,x2 ,x3 ,x4
C (x1 , x2 , x3 , x4 )
3
Yedidia, Message-passing algorithms for inference and optimization,
Journ. of Stat. Phys. vol. 145, no. 4, Nov. 2011.
11/21
Min-Sum Algorithm3
Interference
Alignment via
Message-Passing
M. Guillaud
I
GDL applied to (R, min, +)
I
Example:
Motivation
Communication Problem
Interference Alignment
C (x1 , x2 , x3 , x4 ) = Ca (x1 , x2 , x3 ) + Cb (x2 , x4 ) + Cc (x3 , x4 )
Message-Passing
GDL
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
Numerical Results
Conclusion
I
Min-Sum can solve efficiently
min
x1 ,x2 ,x3 ,x4
C (x1 , x2 , x3 , x4 )
3
Yedidia, Message-passing algorithms for inference and optimization,
Journ. of Stat. Phys. vol. 145, no. 4, Nov. 2011.
11/21
Min-Sum Algorithm3
Interference
Alignment via
Message-Passing
M. Guillaud
I
GDL applied to (R, min, +)
I
Example:
Motivation
Communication Problem
Interference Alignment
C (x1 , x2 , x3 , x4 ) = Ca (x1 , x2 , x3 ) + Cb (x2 , x4 ) + Cc (x3 , x4 )
Message-Passing
GDL
Min-Sum
l
IA via Min-Sum
Implementation challenges
Distributedness
Numerical Results
Conclusion
I
Min-Sum can solve efficiently
min
x1 ,x2 ,x3 ,x4
C (x1 , x2 , x3 , x4 )
3
Yedidia, Message-passing algorithms for inference and optimization,
Journ. of Stat. Phys. vol. 145, no. 4, Nov. 2011.
11/21
Min-Sum Algorithm3
Interference
Alignment via
Message-Passing
M. Guillaud
I
GDL applied to (R, min, +)
I
Example:
Motivation
Communication Problem
Interference Alignment
C (x1 , x2 , x3 , x4 ) = Ca (x1 , x2 , x3 ) + Cb (x2 , x4 ) + Cc (x3 , x4 )
Message-Passing
GDL
Min-Sum
l
IA via Min-Sum
Implementation challenges
Distributedness
Numerical Results
Conclusion
I
Min-Sum can solve efficiently
min
x1 ,x2 ,x3 ,x4
C (x1 , x2 , x3 , x4 )
3
Yedidia, Message-passing algorithms for inference and optimization,
Journ. of Stat. Phys. vol. 145, no. 4, Nov. 2011.
11/21
Interference
Alignment via
Message-Passing
Min-Sum Implementation
I
factor-to-variable update
M. Guillaud
Motivation
Communication Problem
Interference Alignment
Message-Passing
GDL
Min-Sum
IA via Min-Sum
ma→i (xi ) = min [Ca (xi , xj , xk ) + mj→a (xj ) + mk→a (xk )]
xj ,xk
Implementation challenges
Distributedness
Numerical Results
I
variable-to-factor update
mi→a (xi ) =
Conclusion
X
mb→i (xi )
b∈N(i)\a
(illustrations from [Yedidia 2011])
12/21
Interference
Alignment via
Message-Passing
Min-Sum Implementation
I
factor-to-variable update
M. Guillaud
Motivation
Communication Problem
Interference Alignment
Message-Passing
GDL
Min-Sum
IA via Min-Sum
ma→i (xi ) = min [Ca (xi , xj , xk ) + mj→a (xj ) + mk→a (xk )]
xj ,xk
Implementation challenges
Distributedness
Numerical Results
I
variable-to-factor update
mi→a (xi ) =
Conclusion
X
mb→i (xi )
b∈N(i)\a
(illustrations from [Yedidia 2011])
12/21
Interference
Alignment via
Message-Passing
Back to Our Interference Problem...
I
Reformulate the IA equations
U†i Hij Vj = 0 ∀i 6= j
M. Guillaud
⇔
2
X †
Ui Hij Vj = 0
F
i6=j
Motivation
Communication Problem
Interference Alignment
Message-Passing
GDL
m
Min-Sum
IA via Min-Sum
Implementation challenges
K
X
X
i=1
j6=i
Distributedness
trace U†i Hij Vj V†j Hij Ui = 0
Numerical Results
Conclusion
13/21
Interference
Alignment via
Message-Passing
Back to Our Interference Problem...
I
Reformulate the IA equations
U†i Hij Vj = 0 ∀i 6= j
M. Guillaud
2
X †
Ui Hij Vj = 0
⇔
i6=j
F
Motivation
Communication Problem
Interference Alignment
Message-Passing
GDL
m
Min-Sum
IA via Min-Sum
Implementation challenges
K
X
X
i=1
j6=i
K
X
X
trace U†i Hij Vj V†j Hij Ui +
j=1
i6=j
Distributedness
trace U†i Hij Vj V†j Hij Ui = 0
Numerical Results
Conclusion
13/21
Interference
Alignment via
Message-Passing
Back to Our Interference Problem...
I
Reformulate the IA equations
U†i Hij Vj = 0 ∀i 6= j
M. Guillaud
2
X †
Ui Hij Vj = 0
⇔
F
i6=j
Motivation
Communication Problem
Interference Alignment
Message-Passing
GDL
m
Min-Sum
IA via Min-Sum
Implementation challenges
K
X
X
i=1
K
X
X
trace U†i Hij Vj V†j Hij Ui +
j6=i
|
j=1
{z
fi (Ui ,Vj6=i )
}
Distributedness
trace U†i Hij Vj V†j Hij Ui = 0
Conclusion
i6=j
|
Numerical Results
{z
gj (Vj ,Ui6=j )
}
13/21
Interference
Alignment via
Message-Passing
Back to Our Interference Problem...
I
Reformulate the IA equations
U†i Hij Vj = 0 ∀i 6= j
M. Guillaud
2
X †
Ui Hij Vj = 0
⇔
F
i6=j
Motivation
Communication Problem
Interference Alignment
Message-Passing
GDL
m
Min-Sum
IA via Min-Sum
Implementation challenges
K
X
X
i=1
K
X
X
trace U†i Hij Vj V†j Hij Ui +
j6=i
j=1
{z
|
I
}
fi (Ui ,Vj6=i )
Distributedness
trace U†i Hij Vj V†j Hij Ui = 0
Conclusion
i6=j
|
Numerical Results
{z
gj (Vj ,Ui6=j )
}
Solve via Min-Sum
min
Ui=1...K ∈ GM,d ,
Vj=1...K ∈ GN,d
K
K
X
X
gj Vj , {Ui }i6=j
fi Ui , {Vj }j6=i +
i=1
j=1
13/21
Graphical representation of IA on the 3-user IC
Interference
Alignment via
Message-Passing
M. Guillaud
Motivation
Communication Problem
Interference Alignment
Message-Passing
GDL
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
Numerical Results
Conclusion
14/21
Interference
Alignment via
Message-Passing
Challenge #1: Continuity
M. Guillaud
I
Consider one message mfi →Ui :
h
i
X
mfi →Ui (Ui ) = min fi (Ui , Vj6=i ) +
mVj →fi (Vj )
Vj6=i
j6=i
Motivation
Communication Problem
Interference Alignment
Message-Passing
GDL
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
I
This message is a function with argument in GM,d
I
Compute and pass mfi →Ui (Ui ) for each possible value of Ui ?
I
Parameterize the function!
I
We propose (for X ∈ GM,d )
ma→b (X) = trace X† Qa→b X
Numerical Results
Conclusion
−→ ma→b is represented by a single matrix Qa→b
15/21
Interference
Alignment via
Message-Passing
Challenge #1: Continuity
M. Guillaud
I
Consider one message mfi →Ui :
h
i
X
mfi →Ui (Ui ) = min fi (Ui , Vj6=i ) +
mVj →fi (Vj )
Vj6=i
j6=i
Motivation
Communication Problem
Interference Alignment
Message-Passing
GDL
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
I
This message is a function with argument in GM,d
I
Compute and pass mfi →Ui (Ui ) for each possible value of Ui ?
I
Parameterize the function!
I
We propose (for X ∈ GM,d )
ma→b (X) = trace X† Qa→b X
Numerical Results
Conclusion
−→ ma→b is represented by a single matrix Qa→b
15/21
Interference
Alignment via
Message-Passing
Challenge #1: Continuity
M. Guillaud
I
Consider one message mfi →Ui :
h
i
X
mfi →Ui (Ui ) = min fi (Ui , Vj6=i ) +
mVj →fi (Vj )
Vj6=i
j6=i
Motivation
Communication Problem
Interference Alignment
Message-Passing
GDL
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
I
This message is a function with argument in GM,d
I
Compute and pass mfi →Ui (Ui ) for each possible value of Ui ?
I
Parameterize the function!
I
We propose (for X ∈ GM,d )
ma→b (X) = trace X† Qa→b X
Numerical Results
Conclusion
−→ ma→b is represented by a single matrix Qa→b
15/21
Interference
Alignment via
Message-Passing
Challenge #1: Continuity
M. Guillaud
I
Consider one message mfi →Ui :
h
i
X
mfi →Ui (Ui ) = min fi (Ui , Vj6=i ) +
mVj →fi (Vj )
Vj6=i
j6=i
Motivation
Communication Problem
Interference Alignment
Message-Passing
GDL
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
I
This message is a function with argument in GM,d
I
Compute and pass mfi →Ui (Ui ) for each possible value of Ui ?
I
Parameterize the function!
I
We propose (for X ∈ GM,d )
ma→b (X) = trace X† Qa→b X
Numerical Results
Conclusion
−→ ma→b is represented by a single matrix Qa→b
15/21
Interference
Alignment via
Message-Passing
Challenge #1: Continuity
M. Guillaud
I
Consider one message mfi →Ui :
h
i
X
mfi →Ui (Ui ) = min fi (Ui , Vj6=i ) +
mVj →fi (Vj )
Vj6=i
j6=i
Motivation
Communication Problem
Interference Alignment
Message-Passing
GDL
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
I
This message is a function with argument in GM,d
I
Compute and pass mfi →Ui (Ui ) for each possible value of Ui ?
I
Parameterize the function!
I
We propose (for X ∈ GM,d )
ma→b (X) = trace X† Qa→b X
Numerical Results
Conclusion
−→ ma→b is represented by a single matrix Qa→b
15/21
Interference
Alignment via
Message-Passing
Challenge #2: The Importance of being Closed-Form
M. Guillaud
I
Simple variable-to-functions message computation:
X
X
mgj →Ui (Ui )
mUi →fi (Ui ) =
j6=i
⇔
Motivation
Communication Problem
Interference Alignment
Qgj →Ui . . .
QUi →fi =
j6=i
Message-Passing
GDL
Min-Sum
IA via Min-Sum
I
Function-to-variable messages are more tricky:
X
†
†
†
mfi →Ui (Ui )
=
I
I
Distributedness
Numerical Results
min trace Vj Hij Ui Ui Hij + QVj →fi Vj
j6=i
I
Implementation challenges
Vj
Conclusion
Each minimization is an eigenspace problem
Resort to approximation
to express mfi →Ui as
trace X† Qfi →Ui X
Approximation becomes exact in the vicinity of the
solution
16/21
Interference
Alignment via
Message-Passing
Challenge #2: The Importance of being Closed-Form
M. Guillaud
I
Simple variable-to-functions message computation:
X
X
mgj →Ui (Ui )
mUi →fi (Ui ) =
j6=i
⇔
Motivation
Communication Problem
Interference Alignment
Qgj →Ui . . .
QUi →fi =
j6=i
Message-Passing
GDL
Min-Sum
IA via Min-Sum
I
Function-to-variable messages are more tricky:
X
†
†
†
mfi →Ui (Ui )
I
I
Distributedness
Numerical Results
min trace Vj Hij Ui Ui Hij + QVj →fi Vj
=
j6=i
I
Implementation challenges
Vj
Conclusion
Each minimization is an eigenspace problem
Resort to approximation
to express mfi →Ui as
trace X† Qfi →Ui X
Approximation becomes exact in the vicinity of the
solution
16/21
Interference
Alignment via
Message-Passing
Challenge #2: The Importance of being Closed-Form
M. Guillaud
I
Simple variable-to-functions message computation:
X
X
mgj →Ui (Ui )
mUi →fi (Ui ) =
j6=i
⇔
Motivation
Communication Problem
Interference Alignment
Qgj →Ui . . .
QUi →fi =
j6=i
Message-Passing
GDL
Min-Sum
IA via Min-Sum
I
Function-to-variable messages are more tricky:
X
†
†
†
mfi →Ui (Ui )
I
I
Distributedness
Numerical Results
min trace Vj Hij Ui Ui Hij + QVj →fi Vj
=
j6=i
I
Implementation challenges
Vj
Conclusion
Each minimization is an eigenspace problem
Resort to approximation
to express mfi →Ui as
trace X† Qfi →Ui X
Approximation becomes exact in the vicinity of the
solution
16/21
Interference
Alignment via
Message-Passing
Challenge #2: The Importance of being Closed-Form
M. Guillaud
I
Simple variable-to-functions message computation:
X
X
mgj →Ui (Ui )
mUi →fi (Ui ) =
j6=i
⇔
Motivation
Communication Problem
Interference Alignment
Qgj →Ui . . .
QUi →fi =
j6=i
Message-Passing
GDL
Min-Sum
IA via Min-Sum
I
Function-to-variable messages are more tricky:
X
†
†
†
mfi →Ui (Ui )
I
I
Distributedness
Numerical Results
min trace Vj Hij Ui Ui Hij + QVj →fi Vj
=
j6=i
I
Implementation challenges
Vj
Conclusion
Each minimization is an eigenspace problem
Resort to approximation
to express mfi →Ui as
trace X† Qfi →Ui X
Approximation becomes exact in the vicinity of the
solution
16/21
Some Observations on the Distributed Aspect
Interference
Alignment via
Message-Passing
M. Guillaud
Motivation
Communication Problem
I
Factors fi (Ui , Vj6=i ) and gj (Vj , Ui6=j ) rely on local data only
(respectively available at Rx i and Tx j)
Interference Alignment
Message-Passing
GDL
Min-Sum
IA via Min-Sum
I
Mapping of the graph nodes to the physical devices?
Implementation challenges
Distributedness
Numerical Results
Conclusion
17/21
Some Observations on the Distributed Aspect
Interference
Alignment via
Message-Passing
M. Guillaud
Motivation
Communication Problem
I
Factors fi (Ui , Vj6=i ) and gj (Vj , Ui6=j ) rely on local data only
(respectively available at Rx i and Tx j)
Interference Alignment
Message-Passing
GDL
Min-Sum
IA via Min-Sum
I
Mapping of the graph nodes to the physical devices?
Implementation challenges
Distributedness
Numerical Results
Tx 1
Rx 1
Tx 2
Rx 2
Tx 3
Rx 3
Tx K
Rx K
Conclusion
?
↔
17/21
Interference
Alignment via
Message-Passing
Numerical Results
M. Guillaud
Motivation
Communication Problem
Interference Alignment
I
Metric of interest: Residual interference power (leakage)
Message-Passing
GDL
K
K
X
X
fi Ui , {Vj }j6=i +
gj Vj , {Ui }i6=j
i=1
j=1
Min-Sum
IA via Min-Sum
Implementation challenges
Distributedness
Numerical Results
I
Converges reliably to 0 despite lack of optimality proof for
MP when the graph has cycles
I
Convergence speed compares favorably to existing centralized
algorithms (ILM...)
Conclusion
18/21
Interference
Alignment via
Message-Passing
Numerical Results
M. Guillaud
1
10
ILM
MPIA
Motivation
Communication Problem
Interference Alignment
0
10
Message-Passing
GDL
Min-Sum
interference leakage
-1
10
IA via Min-Sum
Implementation challenges
Distributedness
-2
10
Numerical Results
Conclusion
-3
10
-4
10
-5
10
0
10
20
30
40
50
60
iterations
70
80
90
100
Leakage vs. iterations for one realization of Hij , K = 3,
M = N = 4, d = 2.
19/21
Interference
Alignment via
Message-Passing
Numerical Results
M. Guillaud
700
Motivation
Communication Problem
Interference Alignment
600
ILM
MPIA
Message-Passing
GDL
500
Min-Sum
frequency
IA via Min-Sum
Implementation challenges
400
Distributedness
Numerical Results
300
Conclusion
200
100
0
-15
10
-10
-5
10
10
0
10
leakage (log scale)
Empirical distribution of leakage after 100 iterations over random
realizations of Hij , K = 3, M = N = 4, d = 2.
20/21
Conclusion / Open Questions
Interference
Alignment via
Message-Passing
M. Guillaud
Motivation
Communication Problem
Interference Alignment
I
Systematic method to distributedly solve the interference
alignment problem
Message-Passing
GDL
Min-Sum
I
. . . and possibly other simultaneous orthogonalization
problems formulated on the Grassmann manifold
IA via Min-Sum
Implementation challenges
Distributedness
Numerical Results
I
Open issues
I
I
I
Conclusion
Optimality proof is missing
Effect of quantizing the “messages” Qa→b ?
Optimize mapping of the graph nodes to the devices
21/21
Conclusion / Open Questions
Interference
Alignment via
Message-Passing
M. Guillaud
Motivation
Communication Problem
Interference Alignment
I
Systematic method to distributedly solve the interference
alignment problem
Message-Passing
GDL
Min-Sum
I
. . . and possibly other simultaneous orthogonalization
problems formulated on the Grassmann manifold
IA via Min-Sum
Implementation challenges
Distributedness
Numerical Results
I
Open issues
I
I
I
Conclusion
Optimality proof is missing
Effect of quantizing the “messages” Qa→b ?
Optimize mapping of the graph nodes to the devices
21/21
Backup slides
Interference
Alignment via
Message-Passing
M. Guillaud
Backup slides
Backup Slides
22/21
Interference
Alignment via
Message-Passing
About the approximation in mfi →Ui
M. Guillaud
Backup slides
mfi →Ui (Ui )
=
X
j6=i
≈
X
min trace V†j H†ij Ui U†i Hij + QVj →fi Vj
Vj
trace V0j
†
H†ij Ui U†i Hij + QVj →fi V0j
j6=i
where
V0j = arg min trace V†j QVj →fi Vj
Vj
I
Approximation becomes exact in the vicinity of the solution
23/21
Numerical Results: Benchmark
Interference
Alignment via
Message-Passing
M. Guillaud
Benchmark: Iterative leakage minimization (ILM) algorithm from
[Gomadam,Cadambe,Jafar 2011].
I
Iterative
I
Assumes channel reciprocity
I
Over-the-air training phases
I
No formal proof of
convergence
Backup slides
24/21