Maxime GUILLAUD. Huawei Technologies Mathematical and Algorithmic Sciences Laboratory, Paris

Transcription

1 1/21 Maxime GUILLAUD Alignment Huawei Technologies Mathematical and Algorithmic Sciences Laboratory, Paris Optimisation Géométrique sur les Variétés Réunion GdR ISIS, Paris, 21 Nov. 2014

2 2/21 Outline Alignment of the Alignment Problem Message-passing and optimization

3 3/21 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 1 Rx 1 Tx 2 Tx 3 Rx 2 Rx 3 Tx K Rx K Alignment

4 4/21 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 H ij (M N matrix) Tx 1 Rx 1 Tx 2 Rx 2 Tx 3 Rx 3 Alignment K y i = H ij x j j=1 Tx K i = 1... K Rx K Rx i wants to infer x i from y i (all H ij are assumed known at all nodes)

5 4/21 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 H ij (M N matrix) Tx 1 Rx 1 Tx 2 Rx 2 Tx 3 Rx 3 Alignment K y i = H ij x j j=1 Tx K i = 1... K Rx K Rx i wants to infer x i from y i (all H ij are assumed known at all nodes)

6 4/21 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 H ij (M N matrix) Tx 1 Rx 1 Tx 2 Rx 2 Tx 3 Rx 3 Alignment K y i = H ij x j j=1 Tx K i = 1... K Rx K Rx i wants to infer x i from y i (all H ij are assumed known at all nodes)

7 5/21 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 V j. s j is the data to transmit: x j = V j s j (V j G N,d ) At Rx i, signal is projected onto a d-dimensional subspace spanned by a truncated unitary matrix U i ( G M,d ) Alignment K s i = U i y = U i i H ijv j s j j=1 Choose the U i, V j such that U i H ijv j vanishes for i j (interference) but not for i = j

8 5/21 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 V j. s j is the data to transmit: x j = V j s j (V j G N,d ) At Rx i, signal is projected onto a d-dimensional subspace spanned by a truncated unitary matrix U i ( G M,d ) Alignment K s i = U i y = U i i H ijv j s j j=1 Choose the U i, V j such that U i H ijv j vanishes for i j (interference) but not for i = j

9 5/21 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 V j. s j is the data to transmit: x j = V j s j (V j G N,d ) At Rx i, signal is projected onto a d-dimensional subspace spanned by a truncated unitary matrix U i ( G M,d ) Alignment K s i = U i y = U i i H ijv j s j j=1 Choose the U i, V j such that U i H ijv j vanishes for i j (interference) but not for i = j

10 6/21 Mathematical Formulation Given the M N matrices {H ij } i,j {1,...K} and d < M, N find {U i } i {1,...K} in G M,d and {V i } i {1,...K} in in G N,d such that U i H ijv j = 0 i j. Alignment 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.

11 6/21 Mathematical Formulation Given the M N matrices {H ij } i,j {1,...K} and d < M, N find {U i } i {1,...K} in G M,d and {V i } i {1,...K} in in G N,d such that U i H ijv j = 0 i j. Alignment 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.

12 7/21 Matrix Decomposition Formulation Alignment U 1 H H 1K V U H K K1... H KK V K }{{}}{{}}{{} Kd KM KM KN KN Kd =

13 Available Solutions State of the art: No closed-form solution known for general dimensions Iterative solutions exist 1 but have some undesirable properties We propose to use a message-passing (MP) algorithm: distributed solution Alignment 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 Alignment and Applications to Wireless Networks, IEEE Trans. Inf. Theory, Jun /21

14 Available Solutions State of the art: No closed-form solution known for general dimensions Iterative solutions exist 1 but have some undesirable properties We propose to use a message-passing (MP) algorithm: distributed solution Alignment 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 Alignment and Applications to Wireless Networks, IEEE Trans. Inf. Theory, Jun /21

15 Generalized Distributive Law Efficiently compute functions involving many variables that can be decomposed (factorized) into terms involving subsets of the variables Variable Factor dependencies captured by a bipartite graph General MP formulation for an arbitrary commutative semi-ring 2 Alignment 2 Aji and McEliece, The Generalized Distributive Law, IEEE Trans. Inf. Theory, vol. 46, no. 2, Mar /21

16 10/21 Generalized Distributive Law Alignment 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

17 Algorithm 3 applied to (R, min, +) Example: C(x 1, x 2, x 3, x 4 ) = C a (x 1, x 2, x 3 ) + C b (x 2, x 4 ) + C c (x 3, x 4 ) Alignment can solve efficiently min x 1,x 2,x 3,x 4 C(x 1, x 2, x 3, x 4 ) 3 Yedidia, Message-passing algorithms for inference and optimization, Journ. of Stat. Phys. vol. 145, no. 4, Nov /21

18 Algorithm 3 applied to (R, min, +) Example: C(x 1, x 2, x 3, x 4 ) = C a (x 1, x 2, x 3 ) + C b (x 2, x 4 ) + C c (x 3, x 4 ) Alignment can solve efficiently min x 1,x 2,x 3,x 4 C(x 1, x 2, x 3, x 4 ) 3 Yedidia, Message-passing algorithms for inference and optimization, Journ. of Stat. Phys. vol. 145, no. 4, Nov /21

19 Algorithm 3 applied to (R, min, +) Example: C(x 1, x 2, x 3, x 4 ) = C a (x 1, x 2, x 3 ) + C b (x 2, x 4 ) + C c (x 3, x 4 ) Alignment can solve efficiently min x 1,x 2,x 3,x 4 C(x 1, x 2, x 3, x 4 ) 3 Yedidia, Message-passing algorithms for inference and optimization, Journ. of Stat. Phys. vol. 145, no. 4, Nov /21

20 Algorithm 3 applied to (R, min, +) Example: C(x 1, x 2, x 3, x 4 ) = C a (x 1, x 2, x 3 ) + C b (x 2, x 4 ) + C c (x 3, x 4 ) Alignment can solve efficiently min x 1,x 2,x 3,x 4 C(x 1, x 2, x 3, x 4 ) 3 Yedidia, Message-passing algorithms for inference and optimization, Journ. of Stat. Phys. vol. 145, no. 4, Nov /21

21 Implementation factor-to-variable update Alignment m a i (x i ) = min x j,x k [C a (x i, x j, x k ) + m j a (x j ) + m k a (x k )] variable-to-factor update m i a (x i ) = m b i (x i ) b N(i)\a (illustrations from [Yedidia 2011]) 12/21

22 Implementation factor-to-variable update Alignment m a i (x i ) = min x j,x k [C a (x i, x j, x k ) + m j a (x j ) + m k a (x k )] variable-to-factor update m i a (x i ) = m b i (x i ) b N(i)\a (illustrations from [Yedidia 2011]) 12/21

23 13/21 Back to Our Problem... Reformulate the IA equations U i H ijv j = 0 i j U i H 2 ijv j F i j K ( ) trace U i H ijv j V j H iju i = 0 i=1 j i = 0 Alignment

24 13/21 Back to Our Problem... Reformulate the IA equations U i H ijv j = 0 i j U i H 2 ijv j = 0 F i j K ( ) K ( ) trace U i H ijv j V j H iju i + trace U i H ijv j V j H iju i = 0 i=1 j i j=1 i j Alignment

25 13/21 Back to Our Problem... Reformulate the IA equations U i H ijv j = 0 i j i j U i H 2 ijv j = 0 K ( ) K ( ) trace U i H ijv j V j H iju i + trace U i H ijv j V j H iju i = 0 i=1 j i } {{ } f i (U i,v j i ) j=1 i j } {{ } g j (V j,u i j ) F Alignment

26 13/21 Back to Our Problem... Reformulate the IA equations U i H ijv j = 0 i j i j U i H 2 ijv j = 0 K ( ) K ( ) trace U i H ijv j V j H iju i + trace U i H ijv j V j H iju i = 0 i=1 j i } {{ } f i (U i,v j i ) j=1 i j } {{ } g j (V j,u i j ) F Alignment Solve via min U i=1...k G M,d, V j=1...k G N,d K i=1 f i ( U i, {V j } j i ) + K j=1 g j ( V j, {U i } i j )

27 14/21 Graphical representation of IA on the 3-user IC Alignment

28 15/21 Challenge #1: Continuity Consider one message m fi U i : [ m fi U i (U i ) = min f i (U i, V j i ) + ] m Vj f i (V j ) V j i j i This message is a function with argument in G M,d Compute and pass m fi U i (U i ) for each possible value of U i? Alignment Parameterize the function! We propose (for X G M,d ) m a b (X) = trace ( X Q a b X ) m a b is represented by a single matrix Q a b

29 15/21 Challenge #1: Continuity Consider one message m fi U i : [ m fi U i (U i ) = min f i (U i, V j i ) + ] m Vj f i (V j ) V j i j i This message is a function with argument in G M,d Compute and pass m fi U i (U i ) for each possible value of U i? Alignment Parameterize the function! We propose (for X G M,d ) m a b (X) = trace ( X Q a b X ) m a b is represented by a single matrix Q a b

30 15/21 Challenge #1: Continuity Consider one message m fi U i : [ m fi U i (U i ) = min f i (U i, V j i ) + ] m Vj f i (V j ) V j i j i This message is a function with argument in G M,d Compute and pass m fi U i (U i ) for each possible value of U i? Alignment Parameterize the function! We propose (for X G M,d ) m a b (X) = trace ( X Q a b X ) m a b is represented by a single matrix Q a b

31 15/21 Challenge #1: Continuity Consider one message m fi U i : [ m fi U i (U i ) = min f i (U i, V j i ) + ] m Vj f i (V j ) V j i j i This message is a function with argument in G M,d Compute and pass m fi U i (U i ) for each possible value of U i? Alignment Parameterize the function! We propose (for X G M,d ) m a b (X) = trace ( X Q a b X ) m a b is represented by a single matrix Q a b

32 15/21 Challenge #1: Continuity Consider one message m fi U i : [ m fi U i (U i ) = min f i (U i, V j i ) + ] m Vj f i (V j ) V j i j i This message is a function with argument in G M,d Compute and pass m fi U i (U i ) for each possible value of U i? Alignment Parameterize the function! We propose (for X G M,d ) m a b (X) = trace ( X Q a b X ) m a b is represented by a single matrix Q a b

33 16/21 Challenge #2: The Importance of being Closed-Form Simple variable-to-functions message computation: m Ui f i (U i ) = m gj U i (U i ) Q Ui f i = Q gj U i... j i Function-to-variable messages are more tricky: m fi U i (U i ) = j i j i ( [ ] ) min trace V j H ij U i U i H ij + Q Vj f i Vj V j Alignment Each minimization is an eigenspace problem Resort to approximation to express m fi U i as trace ( X Q fi U i X ) Approximation becomes exact in the vicinity of the solution

34 16/21 Challenge #2: The Importance of being Closed-Form Simple variable-to-functions message computation: m Ui f i (U i ) = m gj U i (U i ) Q Ui f i = Q gj U i... j i Function-to-variable messages are more tricky: m fi U i (U i ) = j i j i ( [ ] ) min trace V j H ij U i U i H ij + Q Vj f i Vj V j Alignment Each minimization is an eigenspace problem Resort to approximation to express m fi U i as trace ( X Q fi U i X ) Approximation becomes exact in the vicinity of the solution

35 16/21 Challenge #2: The Importance of being Closed-Form Simple variable-to-functions message computation: m Ui f i (U i ) = m gj U i (U i ) Q Ui f i = Q gj U i... j i Function-to-variable messages are more tricky: m fi U i (U i ) = j i j i ( [ ] ) min trace V j H ij U i U i H ij + Q Vj f i Vj V j Alignment Each minimization is an eigenspace problem Resort to approximation to express m fi U i as trace ( X Q fi U i X ) Approximation becomes exact in the vicinity of the solution

36 16/21 Challenge #2: The Importance of being Closed-Form Simple variable-to-functions message computation: m Ui f i (U i ) = m gj U i (U i ) Q Ui f i = Q gj U i... j i Function-to-variable messages are more tricky: m fi U i (U i ) = j i j i ( [ ] ) min trace V j H ij U i U i H ij + Q Vj f i Vj V j Alignment Each minimization is an eigenspace problem Resort to approximation to express m fi U i as trace ( X Q fi U i X ) Approximation becomes exact in the vicinity of the solution

37 17/21 Some Observations on the Distributed Aspect Factors f i (U i, V j i ) and g j (V j, U i j ) rely on local data only (respectively available at Rx i and Tx j) Mapping of the graph nodes to the physical devices? Alignment

38 17/21 Some Observations on the Distributed Aspect Factors f i (U i, V j i ) and g j (V j, U i j ) rely on local data only (respectively available at Rx i and Tx j) Mapping of the graph nodes to the physical devices? Tx 1 Rx 1 Alignment Tx 2 Rx 2 Tx 3 Rx 3 Tx K Rx K?

39 18/21 Metric of interest: Residual interference power (leakage) K i=1 f i ( U i, {V j } j i ) + K j=1 g j ( V j, {U i } 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...) Alignment

40 19/21 interference leakage ILM MPIA Alignment iterations Leakage vs. iterations for one realization of H ij, K = 3, M = N = 4, d = 2.

41 20/21 frequency ILM MPIA Alignment leakage (log scale) Empirical distribution of leakage after 100 iterations over random realizations of H ij, K = 3, M = N = 4, d = 2.

42 21/21 / 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 Q a b? Optimize mapping of the graph nodes to the devices Alignment

43 21/21 / 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 Q a b? Optimize mapping of the graph nodes to the devices Alignment

44 22/21 Backup slides Backup slides Backup Slides

45 23/21 About the approximation in m fi U i Backup slides m fi U i (U i ) = ( [ ] ) min trace V j H ij U i U i H ij + Q Vj f i Vj V j j i ( [ trace V 0 ] ) j H ij U i U i H ij + Q Vj f i V 0 j j i where V 0 j = arg min V j trace ( V j Q V j f i V j ) Approximation becomes exact in the vicinity of the solution

46 24/21 : Benchmark Benchmark: Iterative leakage minimization (ILM) algorithm from [Gomadam,Cadambe,Jafar 2011]. Backup slides Iterative Assumes channel reciprocity Over-the-air training phases No formal proof of convergence