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 challenges Distributedness Huawei Technologies Numerical Results Mathematical and Algorithmic Sciences Laboratory, Paris Conclusion 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
Interference Outline Alignment via Message-Passing M. Guillaud Motivation Communication Problem Interference Alignment Message-Passing GDL ◮ Motivation of the Interference Alignment Problem Min-Sum IA via Min-Sum Implementation challenges ◮ Message-passing and optimization Distributedness Numerical Results ◮ Numerical Results Conclusion ◮ Conclusion 2/21
Interference Communication Problem Motivation Alignment via Message-Passing M. Guillaud Motivation Communication Problem Interference Alignment Message-Passing ◮ K transmitters (Tx), K receivers (Rx) GDL Tx 1 Rx 1 Min-Sum ◮ Each equipped with an antenna array; IA via Min-Sum Implementation challenges each antenna sends/receives a Distributedness Tx 2 Rx 2 complex scalar Numerical Results Conclusion ◮ Rx i is only interested in the message Tx 3 Rx 3 from the corresponding Tx i ◮ Other transmitters create (unwanted) Tx K Rx K interference 3/21
Interference Communication Problem Motivation Alignment via Message-Passing M. Guillaud Motivation ◮ At each (discrete) time instant, Tx j Communication Problem Rx 1 Tx 1 sends a ( N -dimensional) signal x j , Rx Interference Alignment i receives y i ( M -dimensional) Message-Passing GDL Tx 2 Rx 2 ◮ The gains of the wireless propagation Min-Sum IA via Min-Sum channel between Tx j and Rx i is H ij Implementation challenges Tx 3 Rx 3 Distributedness ( M × N matrix) Numerical Results Conclusion Tx K Rx K K � y i = H ij x j ∀ i = 1 . . . K j =1 ◮ Rx i wants to infer x i from y i (all H ij are assumed known at all nodes) 4/21
Interference Communication Problem Motivation Alignment via Message-Passing M. Guillaud Motivation ◮ At each (discrete) time instant, Tx j Communication Problem Rx 1 Tx 1 sends a ( N -dimensional) signal x j , Rx Interference Alignment i receives y i ( M -dimensional) Message-Passing GDL Tx 2 Rx 2 ◮ The gains of the wireless propagation Min-Sum IA via Min-Sum channel between Tx j and Rx i is H ij Implementation challenges Tx 3 Rx 3 Distributedness ( M × N matrix) Numerical Results Conclusion Tx K Rx K K � y i = H ij x j ∀ i = 1 . . . K j =1 ◮ Rx i wants to infer x i from y i (all H ij are assumed known at all nodes) 4/21
Interference Communication Problem Motivation Alignment via Message-Passing M. Guillaud Motivation ◮ At each (discrete) time instant, Tx j Communication Problem Rx 1 Tx 1 sends a ( N -dimensional) signal x j , Rx Interference Alignment i receives y i ( M -dimensional) Message-Passing GDL Tx 2 Rx 2 ◮ The gains of the wireless propagation Min-Sum IA via Min-Sum channel between Tx j and Rx i is H ij Implementation challenges Tx 3 Rx 3 Distributedness ( M × N matrix) Numerical Results Conclusion Tx K Rx K K � y i = H ij x j ∀ i = 1 . . . K j =1 ◮ Rx i wants to infer x i from y i (all H ij are assumed known at all nodes) 4/21
Interference Inteference Alignment Alignment via Message-Passing Simple, linear transmission scheme that mitigates interference: M. Guillaud ◮ Tx signals are restricted to a d -dimensional subspace of the Motivation N -dimensional space of the antenna array, spanned by a Communication Problem Interference Alignment truncated unitary matrix V j . s j is the data to transmit: Message-Passing GDL x j = V j s j ( V j ∈ G N , d ) Min-Sum IA via Min-Sum Implementation challenges Distributedness Numerical Results ◮ At Rx i , signal is projected onto a d -dimensional subspace Conclusion spanned by a truncated unitary matrix U i ( ∈ G M , d ) K � i = U † U † s ′ i y i = i H ij V j s j j =1 ◮ Choose the U i , V j such that U † i H ij V j vanishes for i � = j (interference) but not for i = j 5/21
Interference Inteference Alignment Alignment via Message-Passing Simple, linear transmission scheme that mitigates interference: M. Guillaud ◮ Tx signals are restricted to a d -dimensional subspace of the Motivation N -dimensional space of the antenna array, spanned by a Communication Problem Interference Alignment truncated unitary matrix V j . s j is the data to transmit: Message-Passing GDL x j = V j s j ( V j ∈ G N , d ) Min-Sum IA via Min-Sum Implementation challenges Distributedness Numerical Results ◮ At Rx i , signal is projected onto a d -dimensional subspace Conclusion spanned by a truncated unitary matrix U i ( ∈ G M , d ) K � i = U † U † s ′ i y i = i H ij V j s j j =1 ◮ Choose the U i , V j such that U † i H ij V j vanishes for i � = j (interference) but not for i = j 5/21
Interference Inteference Alignment Alignment via Message-Passing Simple, linear transmission scheme that mitigates interference: M. Guillaud ◮ Tx signals are restricted to a d -dimensional subspace of the Motivation N -dimensional space of the antenna array, spanned by a Communication Problem Interference Alignment truncated unitary matrix V j . s j is the data to transmit: Message-Passing GDL x j = V j s j ( V j ∈ G N , d ) Min-Sum IA via Min-Sum Implementation challenges Distributedness Numerical Results ◮ At Rx i , signal is projected onto a d -dimensional subspace Conclusion spanned by a truncated unitary matrix U i ( ∈ G M , d ) K � i = U † U † s ′ i y i = i H ij V j s j j =1 ◮ Choose the U i , V j such that U † i H ij V j vanishes for i � = j (interference) but not for i = j 5/21
Interference Mathematical Formulation Alignment via Message-Passing M. Guillaud Motivation Communication Problem Interference Alignment ◮ Given the M × N matrices { H ij } i , j ∈{ 1 ,... K } and d < M , N Message-Passing GDL ◮ find { U i } i ∈{ 1 ,... K } in G M , d and { V i } i ∈{ 1 ,... K } in in G N , d Min-Sum IA via Min-Sum ◮ such that U † Implementation challenges ∀ i � = j . i H ij V j = 0 Distributedness Numerical Results Conclusion ◮ 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
Interference Mathematical Formulation Alignment via Message-Passing M. Guillaud Motivation Communication Problem Interference Alignment ◮ Given the M × N matrices { H ij } i , j ∈{ 1 ,... K } and d < M , N Message-Passing GDL ◮ find { U i } i ∈{ 1 ,... K } in G M , d and { V i } i ∈{ 1 ,... K } in in G N , d Min-Sum IA via Min-Sum ◮ such that U † Implementation challenges ∀ i � = j . i H ij V j = 0 Distributedness Numerical Results Conclusion ◮ 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
Interference Matrix Decomposition Formulation Alignment via Message-Passing M. Guillaud Motivation Communication Problem Interference Alignment Message-Passing GDL Min-Sum IA via Min-Sum U † 0 H 11 H 1 K V 1 ∗ 1 Implementation challenges . . . ... . . ... ... ... Distributedness = . . . . Numerical Results U † 0 H K 1 H KK V K ∗ . . . K Conclusion � �� � � �� � � �� � KM × KN KN × Kd Kd × KM 7/21
Interference Available Solutions Alignment via Message-Passing M. Guillaud Motivation Communication Problem State of the art: Interference Alignment Message-Passing ◮ No closed-form solution known for general dimensions GDL Min-Sum ◮ Iterative solutions exist 1 but have some undesirable properties IA via Min-Sum Implementation challenges Distributedness We propose to use a message-passing (MP) algorithm : Numerical Results ◮ distributed solution Conclusion ◮ use local data ( H ij is known at Rx i and Tx j ) 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 Available Solutions Alignment via Message-Passing M. Guillaud Motivation Communication Problem State of the art: Interference Alignment Message-Passing ◮ No closed-form solution known for general dimensions GDL Min-Sum ◮ Iterative solutions exist 1 but have some undesirable properties IA via Min-Sum Implementation challenges Distributedness We propose to use a message-passing (MP) algorithm : Numerical Results ◮ distributed solution Conclusion ◮ use local data ( H ij is known at Rx i and Tx j ) 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
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