Distributed Projection Approximation Subspace Tracking Based on Consensus Propagation Carolina Reyes, Thibault Hilaire, Christoph F. Mecklenbräuker creyes@nt.tuwien.ac.at, thibault.hilaire@lip6.fr, cfm@nt.tuwien.ac.at Institute of Communications and Radio Frequency Engineering Vienna University of Technology CAMSAP December 16, 2009 INSTITUT FÜR NACHRICHTENTECHNIK UND HOCHFREQUENZTECHNIK
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions Outline Introduction 1 PAST-Consensus Propagation Algorithm 2 Simulation Results 3 Summary and Conclusions 4 2 / 35
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions Problem Statement Finite capacity of communication channel. Bit rate constraints. Sensor network architectures are structured in a centralized/small distributed fashion. Average data collected from the whole network is more important than individual node data. 3 / 35
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions Applications Industrial, building, home system automation. Monitoring (concentrations of chemicals in hydrology, agriculture, pollution control, prediction of avalanches and land slides). Healthcare sensor implantation in human bodies. 4 / 35
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions Projection Approximation Subspace Tracking Algorithm Mathematical model Let x ( t ) ∈ C N be the data vector observed at time t , with r narrow-band signal waves impinging on an array of N sensors x ( t ) = A ( ω ( t ) ) s ( t ) + v ( t ) , 1 1 1 s 1 ( t ) e j ω 1 e j ω 2 e j ω r . . A = , s ( t ) = . . . . . . . . . . s r ( t ) e ( N − 1 ) j ω 1 e ( n − 1 ) j ω 2 e ( n − 1 ) j ω r 5 / 35
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions Projection Approximation Subspace Tracking Algorithm Image source: Euclidean Subspace, Wikipedia 6 / 35
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions Projection Approximation Subspace Tracking Algorithm Cost function Bin Yang [1] proposed to minimize the following cost function J ′ ( W ( t )) = � t � 2 , i = 1 β t − i � � x ( i ) − W ( t ) y ( i ) � by the the approximation y ( i ) = W H ( i − 1 ) x ( i ) . [1] B. Yang, “Projection Approximation Subspace Tracking”, IEEE Trans. Sig. Proc., vol. 43, no. 1, pp. 95-107, 1995. 7 / 35
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions What is Projection Approximation Subspace Tracking? Requires O ( nr ) operations per update. n : Input vector dimension. r : Desired number of eigencomponents. t : Number of snapshots. Constrained to r < n < t . 8 / 35
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions Consensus Propagation 35 30 25 20 23 15 16 18 17 10 10 11 5 0 −5 −5 0 5 10 15 20 25 30 35 Figure: Sensor network with neighborhood N 17 for radius 9 9 / 35
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions Consensus Propagation Figure: Sensor network with neighborhood N 17 for radius 9 10 / 35
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions Consensus Propagation Figure: Sensor network with neighborhood N 17 for radius 9 11 / 35
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions Consensus Propagation Algorithm for n := 1 , 2 , . . . , N do Input { y j ( t − 1 ) , w j } j ∈N n are the pairs sent to node n in step t − 1 � � � � y n ( t ) = y j ( t − 1 ) w j w j � � / j ∈N n j ∈N n Broadcast the pair { y n ( t ) , w n } to all nodes in N n Output: y n ( t ) is the estimation of the average in step t at node n endfor 12 / 35
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions Consensus Propagation Algorithm for n := 1 , 2 , . . . , N do Input { y j ( t − 1 ) , w j } j ∈N n are the pairs sent to node n in step t − 1 � � � � y n ( t ) = y j ( t − 1 ) w j w j � / � j ∈N n j ∈N n endfor 13 / 35
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions Consensus Propagation Algorithm for n := 1 , 2 , . . . , N do Input { y j ( t − 1 ) , w j } j ∈N n are the pairs sent to node n in step t − 1 � � � � y n ( t ) = y j ( t − 1 ) w j w j � / � j ∈N n j ∈N n Broadcast the pair { y n ( t ) , w n } to all nodes in N n , endfor 14 / 35
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions Consensus Propagation Algorithm for n := 1 , 2 , . . . , N do Input { y j ( t − 1 ) , w j } j ∈N n are the pairs sent to node n in step t − 1 � � � � y n ( t ) = y j ( t − 1 ) w j w j � / � j ∈N n j ∈N n Broadcast the pair { y n ( t ) , w n } to all nodes in N n , w n = 1 / � |N n | endfor 15 / 35
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions Consensus Propagation Algorithm for n := 1 , 2 , . . . , N do Input { y j ( t − 1 ) , w j } j ∈N n are the pairs sent to node n in step t − 1 � � � � y n ( t ) = y j ( t − 1 ) w j w j � / � j ∈N n j ∈N n Broadcast the pair { y n ( t ) , w n } to all nodes in N n , w n = 1 / � |N n | endfor 16 / 35
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions Consensus Propagation Algorithm for n := 1 , 2 , . . . , N do Input { y j ( t − 1 ) , w j } j ∈N n are the pairs sent to node n in step t − 1 � � � � y n ( t ) = y j ( t − 1 ) w j w j � / � j ∈N n j ∈N n Broadcast the pair { y n ( t ) , w n } to all nodes in N n , w n = 1 / � |N n | Output: y n ( t ) is the estimation of the average in step t at node n endfor 17 / 35
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions Consensus Propagation Algorithm for n := 1 , 2 , . . . , N do Input { y j ( t − 1 ) , w j } j ∈N n are the pairs sent to node n in step t − 1 � � � � y n ( t ) = y j ( t − 1 ) w j w j � / � j ∈N n j ∈N n Broadcast the pair { y n ( t ) , w n } to all nodes in N n , w n = 1 / � |N n | Output: y n ( t ) is the estimation of the average in step t at node n endfor 18 / 35
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions PAST-Consensus Propagation Algorithm Initialize: β, P 1 ( 0 ) , . . . , P N ( 0 ) , W 1 ( 0 ) , . . . , W N ( 0 ) for t := 1 , 2 , . . . do for n := 1 , 2 , . . . , N do Input: x n ( t ) aggregate x n ( t )= S n x ( t − 1 ) from all nodes ∈N n y n ( t ) = W H n ( t − 1 ) x n ( t ) locally average y n ( t ) h n ( t ) = P n ( t − 1 ) y n ( t ) g n ( t ) = h n ( t ) / [ β + y H n ( t ) h n ( t )] β ( P n ( t − 1 ) − g n ( t ) h H P n ( t ) = 1 n ( t )) e n ( t ) = D n ( x ( t ) − W n ( t − 1 ) y n ( t )) W n ( t ) = W n ( t − 1 ) + e n ( t ) g H n ( t ) broadcast { x n ( t ) , y n ( t ) , w n } to all nodes ∈N n end end 19 / 35
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions PAST-Consensus Propagation Algorithm Initialize: β, P 1 ( 0 ) , . . . , P N ( 0 ) , W 1 ( 0 ) , . . . , W N ( 0 ) for t := 1 , 2 , . . . do for n := 1 , 2 , . . . , N do Input: x n ( t ) end end 20 / 35
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions PAST-Consensus Propagation Algorithm Initialize: β, P 1 ( 0 ) , . . . , P N ( 0 ) , W 1 ( 0 ) , . . . , W N ( 0 ) for t := 1 , 2 , . . . do for n := 1 , 2 , . . . , N do Input: x n ( t ) aggregate x n ( t )= S n x ( t − 1 ) from all nodes ∈N n y n ( t ) = W H n ( t − 1 ) x n ( t ) locally average y n ( t ) end end 21 / 35
Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions PAST-Consensus Propagation Algorithm Initialize: β, P 1 ( 0 ) , . . . , P N ( 0 ) , W 1 ( 0 ) , . . . , W N ( 0 ) for t := 1 , 2 , . . . do for n := 1 , 2 , . . . , N do Input: x n ( t ) aggregate x n ( t )= S n x ( t − 1 ) from all nodes ∈N n y n ( t ) = W H n ( t − 1 ) x n ( t ) locally average y n ( t ) h n ( t ) = P n ( t − 1 ) y n ( t ) g n ( t ) = h n ( t ) / [ β + y H n ( t ) h n ( t )] β ( P n ( t − 1 ) − g n ( t ) h H P n ( t ) = 1 n ( t )) e n ( t ) = D n ( x ( t ) − W n ( t − 1 ) y n ( t )) W n ( t ) = W n ( t − 1 ) + e n ( t ) g H n ( t ) end end 22 / 35
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