Explicit Sensor Network Localization using Semidefinite Programming - PowerPoint PPT Presentation

Explicit Sensor Network Localization using Semidefinite Programming and Clique Reductions Nathan Krislock, Henry Wolkowicz Department of Combinatorics & Optimization University of Waterloo Southern Ontario Numerical Analysis Day May 8, 2009 Nathan Krislock (University of Waterloo) SNL using SDP and Clique Reductions SONAD 2009 1 / 26

Outline Sensor Network Localization (SNL) 1 Introduction Euclidean Distance Matrices and Semidefinite Matrices Clique Reductions of SNL 2 Clique Reductions Completing the EDM Algorithm 3 Clique Unions and Node Absorptions Results Nathan Krislock (University of Waterloo) SNL using SDP and Clique Reductions SONAD 2009 2 / 26

Outline Sensor Network Localization (SNL) 1 Introduction Euclidean Distance Matrices and Semidefinite Matrices Clique Reductions of SNL 2 Clique Reductions Completing the EDM Algorithm 3 Clique Unions and Node Absorptions Results Nathan Krislock (University of Waterloo) SNL using SDP and Clique Reductions SONAD 2009 2 / 26

Outline Sensor Network Localization (SNL) 1 Introduction Euclidean Distance Matrices and Semidefinite Matrices Clique Reductions of SNL 2 Clique Reductions Completing the EDM Algorithm 3 Clique Unions and Node Absorptions Results Nathan Krislock (University of Waterloo) SNL using SDP and Clique Reductions SONAD 2009 2 / 26

Outline Sensor Network Localization (SNL) 1 Introduction Euclidean Distance Matrices and Semidefinite Matrices Clique Reductions of SNL 2 Clique Reductions Completing the EDM Algorithm 3 Clique Unions and Node Absorptions Results Nathan Krislock (University of Waterloo) SNL using SDP and Clique Reductions SONAD 2009 3 / 26

Introduction Motivation Many applications use wireless sensor networks: natural habitat monitoring, weather monitoring, disaster relief operations, . . . The Sensor Network Localization (SNL) Problem Given: Distances between sensors within a fixed radio range Positions of some fixed sensors (called anchors) Goal: Determine locations of sensors Nathan Krislock (University of Waterloo) SNL using SDP and Clique Reductions SONAD 2009 4 / 26

Introduction 10 9 8 7 6 5 4 3 2 1 0 0 2 4 6 8 10 n = 100, m = 9, R = 2 Nathan Krislock (University of Waterloo) SNL using SDP and Clique Reductions SONAD 2009 5 / 26

Introduction Notation p 1 , . . . , p n − m ∈ R r - unknown points (sensors) a 1 , . . . , a m ∈ R r - known points (anchors) anchors also labeled p n − m + 1 , . . . , p n p T   1 � X � . ∈ R n × r . P =  =   . A  p T n r - embedding dimension (usually 2 or 3) Assumptions: n >> m > r R > 0 - radio range Nathan Krislock (University of Waterloo) SNL using SDP and Clique Reductions SONAD 2009 6 / 26

Introduction Graph Realization G = ( V , E , w ) - underlying weighted graph V = { 1 , . . . , n } ( i , j ) ∈ E if w ij = � p i − p j � is known Anchors form clique SNL problem ≡ find realization of graph in R r Euclidean Distance Matrix (EDM) Completion D p ∈ S n - partial EDM: � � p i − p j � 2 if ( i , j ) ∈ E ( D p ) ij = 0 otherwise SNL problem ≡ find EDM completion with embed. dim. = r Nathan Krislock (University of Waterloo) SNL using SDP and Clique Reductions SONAD 2009 7 / 26

Introduction Graph Realization G = ( V , E , w ) - underlying weighted graph V = { 1 , . . . , n } ( i , j ) ∈ E if w ij = � p i − p j � is known Anchors form clique SNL problem ≡ find realization of graph in R r Euclidean Distance Matrix (EDM) Completion D p ∈ S n - partial EDM: � � p i − p j � 2 if ( i , j ) ∈ E ( D p ) ij = 0 otherwise SNL problem ≡ find EDM completion with embed. dim. = r Nathan Krislock (University of Waterloo) SNL using SDP and Clique Reductions SONAD 2009 7 / 26

Outline Sensor Network Localization (SNL) 1 Introduction Euclidean Distance Matrices and Semidefinite Matrices Clique Reductions of SNL 2 Clique Reductions Completing the EDM Algorithm 3 Clique Unions and Node Absorptions Results Nathan Krislock (University of Waterloo) SNL using SDP and Clique Reductions SONAD 2009 8 / 26

EDMs and Semidefinite Matrices Linear Transformation K If D is an EDM with embed. dim. r given by P ∈ R n × r , then: D ij = � p i − p j � 2 p T i p i + p T j p j − 2 p T = i p j � diag ( PP T ) e T + e diag ( PP T ) T − 2 PP T � = ij K ( PP T ) ij = Thus D = K ( Y ) , where: K ( Y ) := diag ( Y ) e T + e diag ( Y ) T − 2 Y Y := PP T and Y = PP T is positive semidefinite ( Y ∈ S n + or Y � 0), rank ( Y ) = r K maps the semidefinite cone, S n + , onto the EDM cone, E n Nathan Krislock (University of Waterloo) SNL using SDP and Clique Reductions SONAD 2009 9 / 26

EDMs and Semidefinite Matrices Linear Transformation K If D is an EDM with embed. dim. r given by P ∈ R n × r , then: D ij = � p i − p j � 2 p T i p i + p T j p j − 2 p T = i p j � diag ( PP T ) e T + e diag ( PP T ) T − 2 PP T � = ij K ( PP T ) ij = Thus D = K ( Y ) , where: K ( Y ) := diag ( Y ) e T + e diag ( Y ) T − 2 Y Y := PP T and Y = PP T is positive semidefinite ( Y ∈ S n + or Y � 0), rank ( Y ) = r K maps the semidefinite cone, S n + , onto the EDM cone, E n Nathan Krislock (University of Waterloo) SNL using SDP and Clique Reductions SONAD 2009 9 / 26

EDMs and Semidefinite Matrices Linear Transformation K If D is an EDM with embed. dim. r given by P ∈ R n × r , then: D ij = � p i − p j � 2 p T i p i + p T j p j − 2 p T = i p j � diag ( PP T ) e T + e diag ( PP T ) T − 2 PP T � = ij K ( PP T ) ij = Thus D = K ( Y ) , where: K ( Y ) := diag ( Y ) e T + e diag ( Y ) T − 2 Y Y := PP T and Y = PP T is positive semidefinite ( Y ∈ S n + or Y � 0), rank ( Y ) = r K maps the semidefinite cone, S n + , onto the EDM cone, E n Nathan Krislock (University of Waterloo) SNL using SDP and Clique Reductions SONAD 2009 9 / 26

EDMs and Semidefinite Matrices Properties of K Define the centered and hollow subspaces S C := { Y ∈ S n : Ye = 0 } S H := { D ∈ S n : diag ( D ) = 0 } and K ( Y ) = diag ( Y ) e T + e diag ( Y ) T − 2 Y = ⇒ range ( K ) = S H n ee T is the For D ∈ S H we have K † ( D ) = − 1 2 JDJ where J := I − 1 orthogonal projection onto { e } ⊥ K and K † are one-to-one and onto: K † ( S H ) = S C and K ( S C ) = S H K † ( E n ) = S n K ( S n + ∩ S C ) = E n + ∩ S C and Nathan Krislock (University of Waterloo) SNL using SDP and Clique Reductions SONAD 2009 10 / 26

EDMs and Semidefinite Matrices Properties of K Define the centered and hollow subspaces S C := { Y ∈ S n : Ye = 0 } S H := { D ∈ S n : diag ( D ) = 0 } and K ( Y ) = diag ( Y ) e T + e diag ( Y ) T − 2 Y = ⇒ range ( K ) = S H n ee T is the For D ∈ S H we have K † ( D ) = − 1 2 JDJ where J := I − 1 orthogonal projection onto { e } ⊥ K and K † are one-to-one and onto: K † ( S H ) = S C and K ( S C ) = S H K † ( E n ) = S n K ( S n + ∩ S C ) = E n + ∩ S C and Nathan Krislock (University of Waterloo) SNL using SDP and Clique Reductions SONAD 2009 10 / 26

EDMs and Semidefinite Matrices Properties of K Define the centered and hollow subspaces S C := { Y ∈ S n : Ye = 0 } S H := { D ∈ S n : diag ( D ) = 0 } and K ( Y ) = diag ( Y ) e T + e diag ( Y ) T − 2 Y = ⇒ range ( K ) = S H n ee T is the For D ∈ S H we have K † ( D ) = − 1 2 JDJ where J := I − 1 orthogonal projection onto { e } ⊥ K and K † are one-to-one and onto: K † ( S H ) = S C and K ( S C ) = S H K † ( E n ) = S n K ( S n + ∩ S C ) = E n + ∩ S C and Nathan Krislock (University of Waterloo) SNL using SDP and Clique Reductions SONAD 2009 10 / 26

EDMs and Semidefinite Matrices Properties of K Define the centered and hollow subspaces S C := { Y ∈ S n : Ye = 0 } S H := { D ∈ S n : diag ( D ) = 0 } and K ( Y ) = diag ( Y ) e T + e diag ( Y ) T − 2 Y = ⇒ range ( K ) = S H n ee T is the For D ∈ S H we have K † ( D ) = − 1 2 JDJ where J := I − 1 orthogonal projection onto { e } ⊥ K and K † are one-to-one and onto: K † ( S H ) = S C and K ( S C ) = S H K † ( E n ) = S n K ( S n + ∩ S C ) = E n + ∩ S C and Nathan Krislock (University of Waterloo) SNL using SDP and Clique Reductions SONAD 2009 10 / 26

EDMs and Semidefinite Matrices Vector Formulation � � p i − p j � 2 = ( D p ) ij , ∀ ( i , j ) ∈ E � Find p 1 , . . . , p n ∈ R r such that � p i − p j � 2 ≥ R 2 , ∀ ( i , j ) / ∈ E Matrix Formulation � W ◦ K ( Y ) = D p � Find P ∈ R n × r such that , where Y = PP T H ◦ K ( Y ) ≥ R 2 Semidefinite Programming (SDP) Relaxation � W ◦ K ( Y ) = D p � Find Y ∈ S n + ∩ S C such that H ◦ K ( Y ) ≥ R 2 Vector/Matrix Formulation is non-convex and NP-HARD SDP Relaxation is convex, but degenerate (strict feasibility fails) Nathan Krislock (University of Waterloo) SNL using SDP and Clique Reductions SONAD 2009 11 / 26