Explicit Sensor Network Localization using Semidefinite Programming and Clique Reductions

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1 Explicit Sensor Network Localization using Semidefinite Programming and Clique Reductions Nathan Krislock and Henry Wolkowicz Dept. of Combinatorics and Optimization University of Waterloo Haifa Matrix Theory Conference May 18-21, 2009

2 Outline 1 The Sensor Network Localization, SNL, Problem Semidefinite Programming (SDP); Facial Structure 2 Single Clique Reduction (Exploit Degeneracy) Two Clique Reduction and EDM Completion Reduction and Completion; Singular Case 3 Clique Unions and Node Absorptions Results (low CPU time; high accuracy)

3 Outline 1 The Sensor Network Localization, SNL, Problem Semidefinite Programming (SDP); Facial Structure 2 Single Clique Reduction (Exploit Degeneracy) Two Clique Reduction and EDM Completion Reduction and Completion; Singular Case 3 Clique Unions and Node Absorptions Results (low CPU time; high accuracy)

4 Outline 1 The Sensor Network Localization, SNL, Problem Semidefinite Programming (SDP); Facial Structure 2 Single Clique Reduction (Exploit Degeneracy) Two Clique Reduction and EDM Completion Reduction and Completion; Singular Case 3 Clique Unions and Node Absorptions Results (low CPU time; high accuracy)

5 3 The Sensor Network Localization, SNL, Problem Semidefinite Programming (SDP); Facial Structure Sensor Network Localization, SNL, Problem Wireless Sensor Network n ad hoc wireless sensors (nodes) to locate in R r, (r is embedding dimension; sensors p i R r, i V := 1,...,n) m of the sensors are anchors, p i, i = n m + 1,...,n) (e.g. using GPS) pairwise distances known within radio range R, D ij = p i p j 2, ij E Current Techniques: Nearest, Weighted, SDP Approx. min Y 0,Y Ω H (K (Y) D) SDP program is: Expensive/low accuracy/implicitly highly degenerate (cliques restrict ranks of feasible Y s)

6 3 The Sensor Network Localization, SNL, Problem Semidefinite Programming (SDP); Facial Structure Sensor Network Localization, SNL, Problem Wireless Sensor Network n ad hoc wireless sensors (nodes) to locate in R r, (r is embedding dimension; sensors p i R r, i V := 1,...,n) m of the sensors are anchors, p i, i = n m + 1,...,n) (e.g. using GPS) pairwise distances known within radio range R, D ij = p i p j 2, ij E Current Techniques: Nearest, Weighted, SDP Approx. min Y 0,Y Ω H (K (Y) D) SDP program is: Expensive/low accuracy/implicitly highly degenerate (cliques restrict ranks of feasible Y s)

7 Applications The Sensor Network Localization, SNL, Problem Semidefinite Programming (SDP); Facial Structure Tracking Humans/Animals/Equipment monitoring: natural habitat; earthquakes and volcanos; weather and ocean currents. military, tracking of goods, vehicle positions, surveillance, (where open-air positioning is not feasible), random deployment in inaccessible terrains or disaster relief operations Future Applications? bicycles, car/computer parts, guns (to prevent theft) students/teenagers (prevent class absence)

8 Applications The Sensor Network Localization, SNL, Problem Semidefinite Programming (SDP); Facial Structure Tracking Humans/Animals/Equipment monitoring: natural habitat; earthquakes and volcanos; weather and ocean currents. military, tracking of goods, vehicle positions, surveillance, (where open-air positioning is not feasible), random deployment in inaccessible terrains or disaster relief operations Future Applications? bicycles, car/computer parts, guns (to prevent theft) students/teenagers (prevent class absence)

9 5 The Sensor Network Localization, SNL, Problem Semidefinite Programming (SDP); Facial Structure Graph G = (V, E,ω) node set V = {1,...,n} edge set (i, j) E ; ω ij = p i p j 2 known approximately The anchors form a clique (complete subgraph) Realization of G in R r : a mapping of node v i p i R r with squared distances given by ω. Corresponding Partial Euclidean Distance Matrix, EDM { d 2 D ij = ij if (i, j) E 0 otherwise, d 2 ij = ω ij are known squared Euclidean distances between sensors p i, p j ; anchors correspond to a clique.

10 5 The Sensor Network Localization, SNL, Problem Semidefinite Programming (SDP); Facial Structure Graph G = (V, E,ω) node set V = {1,...,n} edge set (i, j) E ; ω ij = p i p j 2 known approximately The anchors form a clique (complete subgraph) Realization of G in R r : a mapping of node v i p i R r with squared distances given by ω. Corresponding Partial Euclidean Distance Matrix, EDM { d 2 D ij = ij if (i, j) E 0 otherwise, d 2 ij = ω ij are known squared Euclidean distances between sensors p i, p j ; anchors correspond to a clique.

11 6 The Sensor Network Localization, SNL, Problem Semidefinite Programming (SDP); Facial Structure Sensor Localization Problem/Partial EDM Sensors and Anchors 10 9 Initial position of points sensors anchors sens anch # sensors n = 300, # anchors m = 9, radio range R = 1.2

12 SDP: Cone and Faces The Sensor Network Localization, SNL, Problem Semidefinite Programming (SDP); Facial Structure S+ k, Cone of SDP matrices in Sk inner product A, B = trace AB Löwner (psd) partial order A B, A B Faces of cone K F K is a face of K, denoted F K, if ( x, y K, 1 2 (x + y) F) = (cone {x, y} F). F K, if F K, F K ; F is proper face if {0} F K. F K is exposed if: intersection of K with a hyperplane. face(s) denotes smallest face of K that contains set S.

13 SDP: Cone and Faces The Sensor Network Localization, SNL, Problem Semidefinite Programming (SDP); Facial Structure S+ k, Cone of SDP matrices in Sk inner product A, B = trace AB Löwner (psd) partial order A B, A B Faces of cone K F K is a face of K, denoted F K, if ( x, y K, 1 2 (x + y) F) = (cone {x, y} F). F K, if F K, F K ; F is proper face if {0} F K. F K is exposed if: intersection of K with a hyperplane. face(s) denotes smallest face of K that contains set S.

14 8 Facial Structure of SDP Cone The Sensor Network Localization, SNL, Problem Semidefinite Programming (SDP); Facial Structure S k + is a Facially Exposed Cone All faces are exposed. Faces of S+ k Equivalence to Subspaces of Rk face F S k + is determined by range of any, S relint F, i.e. let S = UΓU T be compact spectral decomposition, with diagonal matrix of eigenvalues Γ S t ++, then: F = US t +U T dim F = t(t + 1)/2. (F associated with R(U))

15 8 Facial Structure of SDP Cone The Sensor Network Localization, SNL, Problem Semidefinite Programming (SDP); Facial Structure S k + is a Facially Exposed Cone All faces are exposed. Faces of S+ k Equivalence to Subspaces of Rk face F S k + is determined by range of any, S relint F, i.e. let S = UΓU T be compact spectral decomposition, with diagonal matrix of eigenvalues Γ S t ++, then: F = US t +U T dim F = t(t + 1)/2. (F associated with R(U))

16 9 The Sensor Network Localization, SNL, Problem Semidefinite Programming (SDP); Facial Structure Matrix with Fixed Principal Submatrix For Y S n, α {1,..., n}: Y[α] denotes principal submatrix formed from rows & cols with indices α. Sets with Fixed Principal Submatrices If α = k and Ȳ Sk, then: S n (α, Ȳ) := { Y S k : Y[α] = Ȳ}, S+ n (α, Ȳ) := { Y S+ k : Y[α] = Ȳ} i.e. the subset of matrices Y S k (Y S+) k with principal submatrix Y[α] fixed to Ȳ.

17 9 The Sensor Network Localization, SNL, Problem Semidefinite Programming (SDP); Facial Structure Matrix with Fixed Principal Submatrix For Y S n, α {1,..., n}: Y[α] denotes principal submatrix formed from rows & cols with indices α. Sets with Fixed Principal Submatrices If α = k and Ȳ Sk, then: S n (α, Ȳ) := { Y S k : Y[α] = Ȳ}, S+ n (α, Ȳ) := { Y S+ k : Y[α] = Ȳ} i.e. the subset of matrices Y S k (Y S+) k with principal submatrix Y[α] fixed to Ȳ.

18 10 The Sensor Network Localization, SNL, Problem Semidefinite Programming (SDP); Facial Structure SNL Connection to SDP (Lin. Trans., Adjoints) v = diag (S) R n, S = Diag (v) S n diag = Diag (adjoint) for B S n, offdiag (B) := B Diag (diag (B)) D = K (B) E n, B = K (D) S n S C Let P T = [ p 1 p 2... p n ] M r n ; B := PP T S n ; D E n be corresponding EDM. (from S n ) K (B) := D e (B) 2B := diag(b) e T + e diag(b) T 2B ( ) n = pi T p i + pj T p j 2pi T p j i,j=1 = ( p i p j 2 2) n i,j=1 = D (to E n ).

19 K : S n + S C E n The Sensor Network Localization, SNL, Problem Semidefinite Programming (SDP); Facial Structure Linear Transformations: D v (B), K (B), T (D) allow: D v (B) := diag(b) v T + v diag(b) T ; D v (y) := yv T + vy T adjoint K (D) = 2(Diag(De) D). K is 1 1, onto between centered and hollow subspaces: S C := {B S n : Be = 0}; S H := {D S n : diag (D) = 0} = R(offDiag ) J := I 1 n eet (orthogonal projection onto M := {e} ); T (D) := 1 2JoffDiag (D)J (= K (D))

20 Properties of Linear Transformations The Sensor Network Localization, SNL, Problem Semidefinite Programming (SDP); Facial Structure K, T, Diag, D e R(K ) = S H ; N (K ) = R (D e ); R(K ) = R(T ) = S C ; N (K ) = N (T ) = R(Diag ); S n = S H R(Diag ) = S C R (D e ). T (E n ) = S n + S C and K (S n + S C ) = E n.

21 13 Single Clique/Facial Reduction Single Clique Reduction (Exploit Degeneracy) Two Clique Reduction and EDM Completion Reduction and Completion; Singular Case D E k, α 1:n, α = k Define E n (α, D) := { D E n : D[α] = D }. Given D; find a corresponding B 0; find the corresponding face; find the corresponding subspace.

22 Single Clique Reduction (Exploit Degeneracy) Two Clique Reduction and EDM Completion Reduction and Completion; Singular Case Basic Theorem for Single Clique/Facial Reduction THEOREM 1: Single Clique Reduction Let D := D[1:k] E k, k < n, with embedding dimension t r, and B := K ( D) = ŪBSŪT B, where ŪB M k t ], ŪT B ŪB = I t, and S S++. t 1 Furthermore, let U B := [ŪB k e M k (t+1), [ ] UB 0 [ ] U U :=, and let V T e M n k+t+1 be 0 I U T e n k orthogonal. Then: face K ( E n (1:k, D) ) ( ) = US+ n k+t+1 U T S C = (UV)S+ n k+t (UV) T Note that we add 1 k e to represent N (K ); then we use V to eliminate e to recover centered face.

23 Single Clique Reduction (Exploit Degeneracy) Two Clique Reduction and EDM Completion Reduction and Completion; Singular Case Positive Integers for Intersecting Cliques Two Intersection Sets, α 1,α 2. α 1 α 2 k 1 k 2 k3 For each clique α = k, we get a corresponding face/subspace (k r matrix) representation. We now see how to handle two cliques, α 1,α 2, that intersect. 15

24 16 Single Clique Reduction (Exploit Degeneracy) Two Clique Reduction and EDM Completion Reduction and Completion; Singular Case Two (Intersecting) Clique Reduction/Subsp. Repres. THEOREM 2: Clique Intersection Reduction/Subsp. Repres. α 1 := 1:( k 1 + k 2 ), α 2 := ( k 1 + 1):( k 1 + k 2 + k 3 ) 1:n, k 1 := α 1 = k 1 + k 2, k 2 := α 2 = k 2 + k 3, k := k 1 + k 2 + k 3. For i = 1, 2, let D i := D[α i ] E k i, with embedding dimension t i, and B i := K ( D i ) = ŪiS i Ūi T, where Ūi M k i t i, ŪT i Ū i = I ti, and S i S t i ++. Furthermore, for i = 1, 2, let 1 U i := [Ūi e ] ki M k i (t i +1), and let Ū M k (t+1) satisfy ([ ]) ([ ]) R(Ū) = R U1 0 R I k1 0, with 0 I k3 0 U ŪT Ū = I t+1 2 cont...

25 Single Clique Reduction (Exploit Degeneracy) Two Clique Reduction and EDM Completion Reduction and Completion; Singular Case Two (Intersecting) Clique Reduction, cont... THEOREM 2 Nonsing. Clique Inters. cont... cont... with ([ ]) ([ ]) R(Ū) = R U1 0 R I k1 0, with 0 I k3 0 U ŪT Ū = I t+1, 2 [Ū ] 0 let U := M n (n k+t+1) and let [ 0 ] I n k U V T e M n k+t+1 be orthogonal. Then U T e 2 i=1 face K ( E n (α i, D i ) ) ( ) = US+ n k+t+1 U T S C = (UV)S+ n k+t (UV) T Basic work: find Ū to repres. inters. of 2 subspaces.

26 Single Clique Reduction (Exploit Degeneracy) Two Clique Reduction and EDM Completion Reduction and Completion; Singular Case Two (Intersecting) Clique Reduction Figure C i C j Completion: missing distances can be recovered if desired. 18

27 19 Single Clique Reduction (Exploit Degeneracy) Two Clique Reduction and EDM Completion Reduction and Completion; Singular Case Two (Intersecting) Clique Explicit Completion COR. Intersection with Embedding Dim. r/completion Hypotheses of Theorem 2 holds. Let D i := D[α i ] E k i, for i = 1, 2, β α 1 α 2,γ := α 1 α 2, D := D[β], B := K ( D), Ū β := Ū(β,:), where Ū M k (t+1) satisfies ] intersection equation of Theorem 2. Let [ V M t+1 Ū T e ŪT e be orthogonal. Let Z := (JŪβ V) B((JŪβ V) ) T. If the embedding dimension for D is r, THEN t = r in Theorem 2, and Z S+ r is the unique solution of the equation (JŪβ V)Z(JŪβ V) T = B, and the exact completion is D[γ] = K ( (Ū V)Z(Ū V) T).

28 Use R as lower bound in singular/nonrigid case. 20 Single Clique Reduction (Exploit Degeneracy) Two Clique Reduction and EDM Completion Reduction and Completion; Singular Case 2 (Inters.) Clique Red. Figure/Singular Case Two (Intersecting) Clique Reduction Figure/Singular Case C i C j

29 Single Clique Reduction (Exploit Degeneracy) Two Clique Reduction and EDM Completion Reduction and Completion; Singular Case Two (Inters.) Clique Explicit Compl.; Sing. Case COR. Clique-Sing.; Intersect. Embedding Dim. r 1 Hypotheses of previous COR holds. For i = 1, 2, let β δ i α i, A i := JŪδ i V, where Ū δi := Ū(δ i,:), and B i := K (D[δ i ]). Let Z S t be a particular solution of the linear systems A 1 ZA T 1 = B 1 A 2 ZA T 2 = B 2.. If the embedding dimension of D[δ i ] is r, for i = 1, 2, but the embedding dimension of D := D[β] is r 1, then the following holds. cont...

30 Single Clique Reduction (Exploit Degeneracy) Two Clique Reduction and EDM Completion Reduction and Completion; Singular Case 2 (Inters.) Clique Expl. Compl.; Degen. cont... COR. Clique-Degen. cont... The following holds: 1 dim N (A i ) = 1, for i = 1, 2. 2 For i = 1, 2, let n i N (A i ), n i 2 = 1, and Z := n 1 n2 T + n 2n1 T. Then, Z is a solution of the linear systems if and only if Z = Z + τ Z, for some τ R 3 There are at most two nonzero solutions, τ 1 and τ 2, for the generalized eigenvalue problem Zv = τ Z v, v 0. Set Z i := Z + 1 τ i Z, for i = 1, 2. Then the exact completion is one of D[γ] { K (Ū V Z i V T Ū T ) : i = 1, 2 }

31 23 Clique Unions and Node Absorptions Results (low CPU time; high accuracy) Initialize C i := { j : (D p ) ij < (R/2) 2}, for i = 1,...,n Iterate Finalize For C i C j r + 1, do Rigid Clique Union For C i N (j) r + 1, do Rigid Node Absorption For C i C j = r, do Non-Rigid Clique Union (lower bnds) For C i N (j) = r, do Non-Rigid Node Absorp. (lower bnds) When a clique containing all anchors, use computed facial representation and positions of anchors to solve for X

32 23 Clique Unions and Node Absorptions Results (low CPU time; high accuracy) Initialize C i := { j : (D p ) ij < (R/2) 2}, for i = 1,...,n Iterate Finalize For C i C j r + 1, do Rigid Clique Union For C i N (j) r + 1, do Rigid Node Absorption For C i C j = r, do Non-Rigid Clique Union (lower bnds) For C i N (j) = r, do Non-Rigid Node Absorp. (lower bnds) When a clique containing all anchors, use computed facial representation and positions of anchors to solve for X

33 23 Clique Unions and Node Absorptions Results (low CPU time; high accuracy) Initialize C i := { j : (D p ) ij < (R/2) 2}, for i = 1,...,n Iterate Finalize For C i C j r + 1, do Rigid Clique Union For C i N (j) r + 1, do Rigid Node Absorption For C i C j = r, do Non-Rigid Clique Union (lower bnds) For C i N (j) = r, do Non-Rigid Node Absorp. (lower bnds) When a clique containing all anchors, use computed facial representation and positions of anchors to solve for X

34 Clique Unions and Node Absorptions Results (low CPU time; high accuracy) Clique Union Node Absorption C i Rigid C i C j j Non-rigid C i Ci C j j

35 Results Clique Unions and Node Absorptions Results (low CPU time; high accuracy) Rigid Clique Union n / R Remaining Cliques n / R CPU Seconds n / R Rigid Clique Union and Node Absorption n / R Remaining Cliques n / R CPU Seconds n / R Max log(error) 25 Max log(error)

36 26 SDP relaxation of SNL is highly (implicitly) degenerate: The feasible set of this SDP is restricted to a low dim. face of the SDP cone, causing the Slater constraint qualification (strict feasibility) to fail We take advantage of this degeneracy by finding explicit representations of intersections of faces of the SDP cone corresponding to unions of intersecting cliques Without using an SDP-solver (eg. SeDuMi or SDPT3), we quickly compute the exact solution to the SDP relaxation (except for round-off error from computing eigenvectors, etc.)

37 26 SDP relaxation of SNL is highly (implicitly) degenerate: The feasible set of this SDP is restricted to a low dim. face of the SDP cone, causing the Slater constraint qualification (strict feasibility) to fail We take advantage of this degeneracy by finding explicit representations of intersections of faces of the SDP cone corresponding to unions of intersecting cliques Without using an SDP-solver (eg. SeDuMi or SDPT3), we quickly compute the exact solution to the SDP relaxation (except for round-off error from computing eigenvectors, etc.)

38 26 SDP relaxation of SNL is highly (implicitly) degenerate: The feasible set of this SDP is restricted to a low dim. face of the SDP cone, causing the Slater constraint qualification (strict feasibility) to fail We take advantage of this degeneracy by finding explicit representations of intersections of faces of the SDP cone corresponding to unions of intersecting cliques Without using an SDP-solver (eg. SeDuMi or SDPT3), we quickly compute the exact solution to the SDP relaxation (except for round-off error from computing eigenvectors, etc.)

39 Thanks for your attention! Explicit Sensor Network Localization using Semidefinite Programming and Clique Reductions Nathan Krislock and Henry Wolkowicz Dept. of Combinatorics and Optimization University of Waterloo Haifa Matrix Theory Conference May 18-21, 2009

Explicit Sensor Network Localization using Semidefinite Programming and Facial Reduction

Explicit Sensor Network Localization using Semidefinite Programming and Facial Reduction Nathan Krislock and Henry Wolkowicz Dept. of Combinatorics and Optimization University of Waterloo INFORMS, San