Interference Alignment via Message-Passing Message-Passing M. - - PowerPoint PPT Presentation

▶
interference alignment via message passing
SMART_READER_LITE
LIVE PREVIEW

Interference Alignment via Message-Passing Message-Passing M. - - PowerPoint PPT Presentation

Feb 16, 2023 225 likes •703 views

Interference Alignment via Interference Alignment via Message-Passing Message-Passing M. Guillaud Motivation Maxime GUILLAUD Communication Problem Interference Alignment Message-Passing GDL Min-Sum IA via Min-Sum Implementation

slide-1
SLIDE 1

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

Interference Alignment via Message-Passing

Maxime GUILLAUD

Huawei Technologies Mathematical and Algorithmic Sciences Laboratory, Paris

maxime.guillaud@huawei.com http://research.mguillaud.net/

Optimisation G´ eom´ etrique sur les Vari´ et´ es R´ eunion GdR ISIS, Paris, 21 Nov. 2014

1/21

slide-2
SLIDE 2

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

Outline

◮ Motivation of the Interference Alignment Problem ◮ Message-passing and optimization ◮ Numerical Results ◮ Conclusion

2/21

slide-3
SLIDE 3

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

Communication Problem Motivation

◮ K transmitters (Tx), K receivers (Rx) ◮ Each equipped with an antenna array;

each antenna sends/receives a complex scalar

◮ Rx i is only interested in the message

from the corresponding Tx i

◮ Other transmitters create (unwanted)

interference

Tx K Tx 2 Tx 3 Rx K Rx 2 Rx 3 Tx 1 Rx 1

3/21

slide-4
SLIDE 4

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

Communication Problem Motivation

◮ At each (discrete) time instant, Tx j

sends a (N-dimensional) signal xj, 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)

Tx K Tx 2 Tx 3 Rx K Rx 2 Rx 3 Tx 1 Rx 1

y i =

K

  • j=1

Hijxj ∀i = 1 . . . K

◮ Rx i wants to infer xi from y i (all Hij are assumed known at

all nodes)

4/21

slide-5
SLIDE 5

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

Communication Problem Motivation

◮ At each (discrete) time instant, Tx j

sends a (N-dimensional) signal xj, 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)

Tx K Tx 2 Tx 3 Rx K Rx 2 Rx 3 Tx 1 Rx 1

y i =

K

  • j=1

Hijxj ∀i = 1 . . . K

◮ Rx i wants to infer xi from y i (all Hij are assumed known at

all nodes)

4/21

slide-6
SLIDE 6

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

Communication Problem Motivation

◮ At each (discrete) time instant, Tx j

sends a (N-dimensional) signal xj, 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)

Tx K Tx 2 Tx 3 Rx K Rx 2 Rx 3 Tx 1 Rx 1

y i =

K

  • j=1

Hijxj ∀i = 1 . . . K

◮ Rx i wants to infer xi from y i (all Hij are assumed known at

all nodes)

4/21

slide-7
SLIDE 7

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

Inteference Alignment

Simple, linear transmission scheme that mitigates interference:

◮ 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. sj is the data to transmit: xj = Vjsj (Vj ∈ GN,d)

◮ At Rx i, signal is projected onto a d-dimensional subspace

spanned by a truncated unitary matrix Ui (∈ GM,d) s′

i = U† i y i = K

  • j=1

U†

i HijVjsj

◮ Choose the Ui, Vj such that U†

i HijVj vanishes for i = j

(interference) but not for i = j

5/21

slide-8
SLIDE 8

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

Inteference Alignment

Simple, linear transmission scheme that mitigates interference:

◮ 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. sj is the data to transmit: xj = Vjsj (Vj ∈ GN,d)

◮ At Rx i, signal is projected onto a d-dimensional subspace

spanned by a truncated unitary matrix Ui (∈ GM,d) s′

i = U† i y i = K

  • j=1

U†

i HijVjsj

◮ Choose the Ui, Vj such that U†

i HijVj vanishes for i = j

(interference) but not for i = j

5/21

slide-9
SLIDE 9

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

Inteference Alignment

Simple, linear transmission scheme that mitigates interference:

◮ 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. sj is the data to transmit: xj = Vjsj (Vj ∈ GN,d)

◮ At Rx i, signal is projected onto a d-dimensional subspace

spanned by a truncated unitary matrix Ui (∈ GM,d) s′

i = U† i y i = K

  • j=1

U†

i HijVjsj

◮ Choose the Ui, Vj such that U†

i HijVj vanishes for i = j

(interference) but not for i = j

5/21

slide-10
SLIDE 10

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

Mathematical Formulation

◮ Given the M × N matrices {Hij}i,j∈{1,...K} and d < M, N ◮ find {Ui}i∈{1,...K} in GM,d and {Vi}i∈{1,...K} in in GN,d ◮ such that U†

i HijVj = 0

∀i = j.

◮ Depending on the relative values of M, N, K and d, the

problem can be trivial, difficult, or provably impossible to solve.

◮ Example of non-trivial case: d = 2, M = N = 4, K = 3.

6/21

slide-11
SLIDE 11

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

Mathematical Formulation

◮ Given the M × N matrices {Hij}i,j∈{1,...K} and d < M, N ◮ find {Ui}i∈{1,...K} in GM,d and {Vi}i∈{1,...K} in in GN,d ◮ such that U†

i HijVj = 0

∀i = j.

◮ Depending on the relative values of M, N, K and d, the

problem can be trivial, difficult, or provably impossible to solve.

◮ Example of non-trivial case: d = 2, M = N = 4, K = 3.

6/21

slide-12
SLIDE 12

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

Matrix Decomposition Formulation

 

U†

1

... U†

K

 

  • Kd×KM

 

H11 . . . H1K . . . ... . . . HK1 . . . HKK

 

  • KM×KN

 

V1 ... VK

 

  • KN×Kd

=  

∗ ... ∗

 

7/21

slide-13
SLIDE 13

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

Available Solutions

State of the art:

◮ No closed-form solution known for general dimensions ◮ Iterative solutions exist1 but have some undesirable properties

We propose to use a message-passing (MP) algorithm:

◮ distributed solution ◮ use local data (Hij is known at Rx i and Tx j)

  • 1K. 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

slide-14
SLIDE 14

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

Available Solutions

State of the art:

◮ No closed-form solution known for general dimensions ◮ Iterative solutions exist1 but have some undesirable properties

We propose to use a message-passing (MP) algorithm:

◮ distributed solution ◮ use local data (Hij is known at Rx i and Tx j)

  • 1K. 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

slide-15
SLIDE 15

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

Generalized Distributive Law

◮ Efficiently compute functions involving many variables that

can be decomposed (factorized) into terms involving subsets

  • f the variables

◮ Variable–Factor dependencies captured by a bipartite graph ◮ General MP formulation for an arbitrary commutative

semi-ring2

2Aji and McEliece, The Generalized Distributive Law, IEEE Trans. Inf.

Theory, vol. 46, no. 2, Mar. 2000.

9/21

slide-16
SLIDE 16

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

Generalized Distributive Law

◮ Algorithm operates by “propagating” messages on the graph ◮ If the graph has no cycle, computation is exact and

terminates in finite number of steps

◮ Otherwise, iterate and hope that it converges to something

reasonable

◮ BCJR decoder, Viterbi, FFT algorithms as special cases

10/21

slide-17
SLIDE 17

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

Min-Sum Algorithm3

◮ GDL applied to (R, min, +) ◮ Example:

C(x1, x2, x3, x4) = Ca(x1, x2, x3) + Cb(x2, x4) + Cc(x3, x4)

◮ Min-Sum can solve efficiently

min

x1,x2,x3,x4 C(x1, x2, x3, x4)

3Yedidia, Message-passing algorithms for inference and optimization,

  • Journ. of Stat. Phys. vol. 145, no. 4, Nov. 2011.

11/21

slide-18
SLIDE 18

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

Min-Sum Algorithm3

◮ GDL applied to (R, min, +) ◮ Example:

C(x1, x2, x3, x4) = Ca(x1, x2, x3) + Cb(x2, x4) + Cc(x3, x4)

◮ Min-Sum can solve efficiently

min

x1,x2,x3,x4 C(x1, x2, x3, x4)

3Yedidia, Message-passing algorithms for inference and optimization,

  • Journ. of Stat. Phys. vol. 145, no. 4, Nov. 2011.

11/21

slide-19
SLIDE 19

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

Min-Sum Algorithm3

◮ GDL applied to (R, min, +) ◮ Example:

C(x1, x2, x3, x4) = Ca(x1, x2, x3) + Cb(x2, x4) + Cc(x3, x4)

  • ◮ Min-Sum can solve efficiently

min

x1,x2,x3,x4 C(x1, x2, x3, x4)

3Yedidia, Message-passing algorithms for inference and optimization,

  • Journ. of Stat. Phys. vol. 145, no. 4, Nov. 2011.

11/21

slide-20
SLIDE 20

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

Min-Sum Algorithm3

◮ GDL applied to (R, min, +) ◮ Example:

C(x1, x2, x3, x4) = Ca(x1, x2, x3) + Cb(x2, x4) + Cc(x3, x4)

  • ◮ Min-Sum can solve efficiently

min

x1,x2,x3,x4 C(x1, x2, x3, x4)

3Yedidia, Message-passing algorithms for inference and optimization,

  • Journ. of Stat. Phys. vol. 145, no. 4, Nov. 2011.

11/21

slide-21
SLIDE 21

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

Min-Sum Implementation

◮ factor-to-variable update

ma→i(xi) = min

xj,xk [Ca(xi, xj, xk) + mj→a(xj) + mk→a(xk)]

◮ variable-to-factor update

mi→a(xi) =

  • b∈N(i)\a

mb→i(xi)

(illustrations from [Yedidia 2011]) 12/21

slide-22
SLIDE 22

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

Min-Sum Implementation

◮ factor-to-variable update

ma→i(xi) = min

xj,xk [Ca(xi, xj, xk) + mj→a(xj) + mk→a(xk)]

◮ variable-to-factor update

mi→a(xi) =

  • b∈N(i)\a

mb→i(xi)

(illustrations from [Yedidia 2011]) 12/21

slide-23
SLIDE 23

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

Back to Our Interference Problem...

◮ Reformulate the IA equations

U†

i HijVj = 0

∀i = j ⇔

  • i=j
  • U†

i HijVj

  • 2

F = 0

  • K
  • i=1
  • j=i

trace U†

i HijVjV† j HijUi

  • = 0

13/21

slide-24
SLIDE 24

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

Back to Our Interference Problem...

◮ Reformulate the IA equations

U†

i HijVj = 0

∀i = j ⇔

  • i=j
  • U†

i HijVj

  • 2

F = 0

  • K
  • i=1
  • j=i

trace U†

i HijVjV† j HijUi

  • +

K

  • j=1
  • i=j

trace U†

i HijVjV† j HijUi

  • = 0

13/21

slide-25
SLIDE 25

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

Back to Our Interference Problem...

◮ Reformulate the IA equations

U†

i HijVj = 0

∀i = j ⇔

  • i=j
  • U†

i HijVj

  • 2

F = 0

  • K
  • i=1
  • j=i

trace U†

i HijVjV† j HijUi

  • fi (Ui ,Vj=i )

+

K

  • j=1
  • i=j

trace U†

i HijVjV† j HijUi

  • gj (Vj ,Ui=j )

= 0

13/21

slide-26
SLIDE 26

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

Back to Our Interference Problem...

◮ Reformulate the IA equations

U†

i HijVj = 0

∀i = j ⇔

  • i=j
  • U†

i HijVj

  • 2

F = 0

  • K
  • i=1
  • j=i

trace U†

i HijVjV† j HijUi

  • fi (Ui ,Vj=i )

+

K

  • j=1
  • i=j

trace U†

i HijVjV† j HijUi

  • gj (Vj ,Ui=j )

= 0 ◮ Solve via Min-Sum

min

Ui=1...K ∈ GM,d, Vj=1...K ∈ GN,d

K

  • i=1

fi

  • Ui, {Vj}j=i
  • +

K

  • j=1

gj

  • Vj, {Ui}i=j
  • 13/21
slide-27
SLIDE 27

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

Graphical representation of IA on the 3-user IC

14/21

slide-28
SLIDE 28

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

Challenge #1: Continuity

◮ Consider one message mfi→Ui:

mfi→Ui(Ui) = min

Vj=i

  • fi(Ui, Vj=i) +
  • j=i

mVj→fi(Vj)

  • ◮ This message is a function with argument in GM,d

◮ Compute and pass mfi→Ui(Ui) for each possible value of Ui ? ◮ Parameterize the function! ◮ We propose (for X ∈ GM,d)

ma→b(X) = trace

  • X†Qa→bX
  • −

→ ma→b is represented by a single matrix Qa→b

15/21

slide-29
SLIDE 29

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

Challenge #1: Continuity

◮ Consider one message mfi→Ui:

mfi→Ui(Ui) = min

Vj=i

  • fi(Ui, Vj=i) +
  • j=i

mVj→fi(Vj)

  • ◮ This message is a function with argument in GM,d

◮ Compute and pass mfi→Ui(Ui) for each possible value of Ui ? ◮ Parameterize the function! ◮ We propose (for X ∈ GM,d)

ma→b(X) = trace

  • X†Qa→bX
  • −

→ ma→b is represented by a single matrix Qa→b

15/21

slide-30
SLIDE 30

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

Challenge #1: Continuity

◮ Consider one message mfi→Ui:

mfi→Ui(Ui) = min

Vj=i

  • fi(Ui, Vj=i) +
  • j=i

mVj→fi(Vj)

  • ◮ This message is a function with argument in GM,d

◮ Compute and pass mfi→Ui(Ui) for each possible value of Ui ? ◮ Parameterize the function! ◮ We propose (for X ∈ GM,d)

ma→b(X) = trace

  • X†Qa→bX
  • −

→ ma→b is represented by a single matrix Qa→b

15/21

slide-31
SLIDE 31

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

Challenge #1: Continuity

◮ Consider one message mfi→Ui:

mfi→Ui(Ui) = min

Vj=i

  • fi(Ui, Vj=i) +
  • j=i

mVj→fi(Vj)

  • ◮ This message is a function with argument in GM,d

◮ Compute and pass mfi→Ui(Ui) for each possible value of Ui ? ◮ Parameterize the function! ◮ We propose (for X ∈ GM,d)

ma→b(X) = trace

  • X†Qa→bX
  • −

→ ma→b is represented by a single matrix Qa→b

15/21

slide-32
SLIDE 32

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

Challenge #1: Continuity

◮ Consider one message mfi→Ui:

mfi→Ui(Ui) = min

Vj=i

  • fi(Ui, Vj=i) +
  • j=i

mVj→fi(Vj)

  • ◮ This message is a function with argument in GM,d

◮ Compute and pass mfi→Ui(Ui) for each possible value of Ui ? ◮ Parameterize the function! ◮ We propose (for X ∈ GM,d)

ma→b(X) = trace

  • X†Qa→bX
  • −

→ ma→b is represented by a single matrix Qa→b

15/21

slide-33
SLIDE 33

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

Challenge #2: The Importance of being Closed-Form

◮ Simple variable-to-functions message computation: mUi →fi (Ui) =

  • j=i

mgj →Ui (Ui) ⇔ QUi →fi =

  • j=i

Qgj →Ui . . . ◮ Function-to-variable messages are more tricky: mfi →Ui (Ui) =

  • j=i

min

Vj

trace V†

j

  • H†

ijUiU† i Hij + QVj →fi

  • Vj
  • ◮ Each minimization is an eigenspace problem

◮ Resort to approximation to express mfi→Ui as

trace

  • X†Qfi→UiX
  • ◮ Approximation becomes exact in the vicinity of the

solution

16/21

slide-34
SLIDE 34

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

Challenge #2: The Importance of being Closed-Form

◮ Simple variable-to-functions message computation: mUi →fi (Ui) =

  • j=i

mgj →Ui (Ui) ⇔ QUi →fi =

  • j=i

Qgj →Ui . . . ◮ Function-to-variable messages are more tricky: mfi →Ui (Ui) =

  • j=i

min

Vj

trace V†

j

  • H†

ijUiU† i Hij + QVj →fi

  • Vj
  • ◮ Each minimization is an eigenspace problem

◮ Resort to approximation to express mfi→Ui as

trace

  • X†Qfi→UiX
  • ◮ Approximation becomes exact in the vicinity of the

solution

16/21

slide-35
SLIDE 35

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

Challenge #2: The Importance of being Closed-Form

◮ Simple variable-to-functions message computation: mUi →fi (Ui) =

  • j=i

mgj →Ui (Ui) ⇔ QUi →fi =

  • j=i

Qgj →Ui . . . ◮ Function-to-variable messages are more tricky: mfi →Ui (Ui) =

  • j=i

min

Vj

trace V†

j

  • H†

ijUiU† i Hij + QVj →fi

  • Vj
  • ◮ Each minimization is an eigenspace problem

◮ Resort to approximation to express mfi→Ui as

trace

  • X†Qfi→UiX
  • ◮ Approximation becomes exact in the vicinity of the

solution

16/21

slide-36
SLIDE 36

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

Challenge #2: The Importance of being Closed-Form

◮ Simple variable-to-functions message computation: mUi →fi (Ui) =

  • j=i

mgj →Ui (Ui) ⇔ QUi →fi =

  • j=i

Qgj →Ui . . . ◮ Function-to-variable messages are more tricky: mfi →Ui (Ui) =

  • j=i

min

Vj

trace V†

j

  • H†

ijUiU† i Hij + QVj →fi

  • Vj
  • ◮ Each minimization is an eigenspace problem

◮ Resort to approximation to express mfi→Ui as

trace

  • X†Qfi→UiX
  • ◮ Approximation becomes exact in the vicinity of the

solution

16/21

slide-37
SLIDE 37

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

Some Observations on the Distributed Aspect

◮ Factors fi(Ui, Vj=i) and gj(Vj, Ui=j) rely on local data only

(respectively available at Rx i and Tx j)

◮ Mapping of the graph nodes to the physical devices?

17/21

slide-38
SLIDE 38

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

Some Observations on the Distributed Aspect

◮ Factors fi(Ui, Vj=i) and gj(Vj, Ui=j) rely on local data only

(respectively available at Rx i and Tx j)

◮ Mapping of the graph nodes to the physical devices?

Tx K Tx 2 Tx 3 Rx K Rx 2 Rx 3 Tx 1 Rx 1

?

↔

17/21

slide-39
SLIDE 39

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

Numerical Results

◮ Metric of interest: Residual interference power (leakage)

K

  • i=1

fi

  • Ui, {Vj}j=i
  • +

K

  • j=1

gj

  • Vj, {Ui}i=j
  • ◮ Converges reliably to 0 despite lack of optimality proof for

MP when the graph has cycles

◮ Convergence speed compares favorably to existing centralized

algorithms (ILM...)

18/21

slide-40
SLIDE 40

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

Numerical Results

10 20 30 40 50 60 70 80 90 100 10

  • 5

10

  • 4

10

  • 3

10

  • 2

10

  • 1

10 10

1

iterations interference leakage ILM MPIA

Leakage vs. iterations for one realization of Hij, K = 3, M = N = 4, d = 2.

19/21

slide-41
SLIDE 41

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

Numerical Results

10

  • 15

10

  • 10

10

  • 5

10 100 200 300 400 500 600 700 leakage (log scale) frequency ILM MPIA

Empirical distribution of leakage after 100 iterations over random realizations of Hij, K = 3, M = N = 4, d = 2.

20/21

slide-42
SLIDE 42

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

Conclusion / Open Questions

◮ Systematic method to distributedly solve the interference

alignment problem

◮ . . . and possibly other simultaneous orthogonalization

problems formulated on the Grassmann manifold

◮ Open issues

◮ Optimality proof is missing ◮ Effect of quantizing the “messages” Qa→b ? ◮ Optimize mapping of the graph nodes to the devices 21/21

slide-43
SLIDE 43

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

Conclusion / Open Questions

◮ Systematic method to distributedly solve the interference

alignment problem

◮ . . . and possibly other simultaneous orthogonalization

problems formulated on the Grassmann manifold

◮ Open issues

◮ Optimality proof is missing ◮ Effect of quantizing the “messages” Qa→b ? ◮ Optimize mapping of the graph nodes to the devices 21/21

slide-44
SLIDE 44

Interference Alignment via Message-Passing

  • M. Guillaud

Backup slides

Backup slides

Backup Slides

22/21

slide-45
SLIDE 45

Interference Alignment via Message-Passing

  • M. Guillaud

Backup slides

About the approximation in mfi→Ui

mfi →Ui (Ui) =

  • j=i

min

Vj

trace V†

j

  • H†

ijUiU† i Hij + QVj →fi

  • Vj
  • ≈
  • j=i

trace

  • V0

j †

H†

ijUiU† i Hij + QVj →fi

  • V0

j

  • where

V0

j = arg min Vj

trace V†

j QVj →fi Vj

  • ◮ Approximation becomes exact in the vicinity of the solution

23/21

slide-46
SLIDE 46

Interference Alignment via Message-Passing

  • M. Guillaud

Backup slides

Numerical Results: Benchmark

Benchmark: Iterative leakage minimization (ILM) algorithm from [Gomadam,Cadambe,Jafar 2011].

◮ Iterative ◮ Assumes channel reciprocity ◮ Over-the-air training phases ◮ No formal proof of

convergence

24/21