On SDP and ESDP Relaxation of Sensor Network Localization

Transcription

1 On SDP and ESDP Relaxation of Sensor Network Localization Paul Tseng Mathematics, University of Washington Seattle IASI, Roma September 16, 2008 (joint work with Ting Kei Pong)

2 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 1 Talk Outline Sensor network localization SDP, ESDP relaxations: formulation and properties

3 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 1 Talk Outline Sensor network localization SDP, ESDP relaxations: formulation and properties A robust version of ESDP to handle noises BCGD-barrier method

4 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 1 Talk Outline Sensor network localization SDP, ESDP relaxations: formulation and properties A robust version of ESDP to handle noises BCGD-barrier method Numerical simulations Conclusion & Ongoing work

5 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 2 Sensor Network Localization Basic Problem: n pts in R 2. Know last n m pts ( anchors ) x m+1,..., x n pairs of neighboring pts and Eucl. dist. estimate for d ij 0 (i, j) A with A {(i, j) : 1 i < j n}. Estimate first m pts ( sensors ).

6 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 2 Sensor Network Localization Basic Problem: n pts in R 2. Know last n m pts ( anchors ) x m+1,..., x n pairs of neighboring pts and Eucl. dist. estimate for d ij 0 (i, j) A with A {(i, j) : 1 i < j n}. Estimate first m pts ( sensors ). History?... Graph realization, position estimation in wireless sensor network,

7 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 3 Optimization Problem Formulation υ opt := min x 1,...,x m xi x j 2 d 2 ij (i,j) A

8 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 3 Optimization Problem Formulation υ opt := min x 1,...,x m xi x j 2 d 2 ij (i,j) A Objective function is nonconvex. m can be large (m > 1000)... Problem is NP-hard (reduction from PARTITION)... Use a convex (SDP, SOCP) relaxation. Low soln accuracy OK. Distributed methods.

9 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 4 SDP Relaxation Let X := [x 1 x m ]. Z = [ X I ] T [ X I ] Z = [ Y X T X I ] 0, rankz = 2

10 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 4 SDP Relaxation Let X := [x 1 x m ]. Z = [ X I ] T [ X I ] Z = [ Y X T X I ] 0, rankz = 2 SDP relaxation (Biswas,Ye 03): υ sdp := min Z + (i,j) A,j>m (i,j) A,j m s.t. Z = [ Y X T X Yii 2x T j x i + x j 2 d 2 ij Y ii 2Y ij + Y jj d 2 ij I ] 0 Adding the nonconvex constraint rankz = 2 yields original problem. But SDP relaxation is still expensive to solve for m large..

11 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 5 ESDP Relaxation ESDP relaxation (Wang, Zheng, Boyd, Ye 06): υ esdp := min Z + (i,j) A,j>m (i,j) A,j m [ ] Y X T s.t. Z = X I Y ii Y ij x T i Y ij Y jj x T j [ x i x j ] I Yii x T i 0 I x i Y ii 2x T j x i + x j 2 d 2 ij Y ii 2Y ij + Y jj d 2 ij 0 i m (i, j) A, j m 0 υ esdp υ sdp υ opt. In simulation, ESDP is nearly as strong as SDP, and solvable much faster by IP method.

12 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 6 An Example n = 3, m = 1, d 12 = d 13 = 2 Problem: 0 = min x 1 R 2 x 1 (1, 0) x 1 ( 1, 0) 2 4

13 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 7 SDP/ESDP Relaxation: 0 = min x 1 =(α,β) R 2 Y 11 R s.t. Y 11 2α 3 + Y α 3 Y 11 α β α 1 0 β If solve SDP/ESDP by IP method, then likely get analy. center.

14 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 8 Properties of SDP & ESDP Relaxations Assume each i m is conn. to some j > m in the graph ({1,..., n}, A). Fact 0: Sol(SDP) and Sol(ESDP) are nonempty, closed, convex, bounded. If d ij = x true i x true j (i, j) A noiseless case (x true i = x i i > m), then υ opt = υ sdp = υ esdp = 0 and Z true := [ X true I ] T [ X true I ] is a soln of SDP and ESDP (i.e., Z true Sol(SDP) Sol(ESDP)).

15 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 9 Let tr i [Z] := Y ii x i 2, i = 1,..., m. ith trace Fact 1 (Biswas,Ye 03, T 07, Wang et al 06): For each i, tr i [Z] = 0 Z ri(sol(esdp)) = x i is invariant over Sol(ESDP) (so x i = x true i in noiseless case) Still true with ESDP changed to SDP.

16 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 9 Let tr i [Z] := Y ii x i 2, i = 1,..., m. ith trace Fact 1 (Biswas,Ye 03, T 07, Wang et al 06): For each i, tr i [Z] = 0 Z ri(sol(esdp)) = x i is invariant over Sol(ESDP) (so x i = x true i in noiseless case) Still true with ESDP changed to SDP. Fact 2 (Pong, T 08): Suppose υ opt = 0. For each i, tr i [Z] = 0 Z Sol(ESDP) x i is invariant over Sol(ESDP). Proof is by induction, starting from sensors that neighbor anchors. (Q: True for SDP?)

17 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 10 In practice, there are measurement noises: d 2 ij = x true i x true j 2 + δ ij (i, j) A. When δ := (δ ij ) (i,j) A 0, does tr i [Z] = 0 (with Z ri(sol(esdp))) imply x i x true i?

18 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 10 In practice, there are measurement noises: d 2 ij = x true i x true j 2 + δ ij (i, j) A. When δ := (δ ij ) (i,j) A 0, does tr i [Z] = 0 (with Z ri(sol(esdp))) imply x i x true i? No!.. Fact 3 (Pong, T 08): For δ 0 and for each i, tr i [Z] = 0 Z ri(sol(esdp)) x i x true i. Still true with ESDP changed to SDP. Proof is by counter-example.

19 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 11 An example of sensitivity of ESDP solns to measurement noise: Input distance data: ɛ > 0 d 12 = 4 + (1 ɛ) 2, d 13 = 1 + ɛ, d 14 = 1 ɛ, d 25 = d 26 = 2; m = 2, n = 6. Thus, even when Z Sol(ESDP) is unique, tr i [Z] = 0 fails to certify accuracy of x i in the noisy case!

20 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 12 Robust ESDP Fix any ρ ij > δ ij (i, j) A (ρ > δ ). Let Sol(ρESDP) denote the set of Z = [ Y X T X I ] satisfying Y ii Y ij x T i Y ij Y jj x T j 0 (i, j) A, j m x i [ x j I ] Yii x T i 0 i m x i I Y ii 2x T j x i + x j 2 d 2 ij ρ ij (i, j) A, j > m Y ii 2Y ij + Y jj d 2 ij ρ ij (i, j) A, j m Note: Z true = [ X true I ] T [ X true I ] Sol(ρESDP).

21 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 13 Let Z ρ,δ := arg min Z Sol(ρESDP) (i,j) A,j m i m ln det [ Yii x ln det T i x i I Y ii Y ij x T i Y ij Y jj x T j x i x j I ]

22 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 13 Let Z ρ,δ := arg min Z Sol(ρESDP) (i,j) A,j m i m ln det [ Yii x ln det T i x i I Y ii Y ij x T i Y ij Y jj x T j x i x j I ] Fact 4 (Pong, T 08): η > 0 and ρ > 0 such that for each i, tr i [Z ρ,δ ] < η δ < ρ ρe = lim δ <ρ 0 xρ,δ i = x true i tr i [Z ρ,δ ] > η 10 δ < ρ ρe = x i not invar. over Sol(ESDP) when δ = 0 Moreover, x ρ,δ i x true i 2 A + m tr i [Z ρ,δ ] δ < ρ.

23 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 14 BCGD-Barrier Method Compute Z ρ,δ? Let h a (t) := 1 2 (t a) ( t a)2 + and f µ (Z) := (i,j) A,j>m + µ (i,j) A,j m (i,j) A,j m µ i m h ρij (Y ii 2x T j x i + x j 2 d 2 ij) h ρij (Y ii 2Y ij + Y jj d 2 ij) ln det [ Yii x ln det T i x i I Y ii Y ij x T i Y ij Y jj x T j x i x j I ]

24 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 15 For each (i, j) A with j > m (resp. j m), h ρij (Y ii 2x T j x i + x j 2 d 2 ij ) = 0 Y ii 2x T j x i + x j 2 d 2 ij ρ ij (resp. h ρij (Y ii 2Y ij + Y jj d 2 ij ) = 0 Y ii 2Y ij + Y jj d 2 ij ρ ij). f µ is partially separable, strictly convex & diff. on its domain. For each fixed ρ > δ, argminf µ Z ρ,δ as µ 0. In the noiseless case (δ = 0), if we set ρ = 0, then argminf µ Z as µ 0, for some Z ri (Sol (ESDP)).

25 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 15 For each (i, j) A with j > m (resp. j m), h ρij (Y ii 2x T j x i + x j 2 d 2 ij ) = 0 Y ii 2x T j x i + x j 2 d 2 ij ρ ij (resp. h ρij (Y ii 2Y ij + Y jj d 2 ij ) = 0 Y ii 2Y ij + Y jj d 2 ij ρ ij). f µ is partially separable, strictly convex & diff. on its domain. For each fixed ρ > δ, argminf µ Z ρ,δ as µ 0. In the noiseless case (δ = 0), if we set ρ = 0, then argminf µ Z as µ 0, for some Z ri (Sol (ESDP)). Idea: Approx. min f µ by block-coordinate gradient descent (BCGD). (T, Yun 06)

26 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 16 BCGD-Barrier Method: Given Z in domf µ, compute gradient Zi f µ of f µ w.r.t. Z i := {x i, Y ii, Y ij : (i, j) A} for each i. If Zi f µ max{µ, 10 6 } for some i, update Z i by moving along the ( ) 1 Newton direction 2 Z i Z i f µ Zi f µ with Armijo stepsize rule. Decrease µ when Zi f µ < max{µ, 10 6 } i. µ initial = 100, µ final = Decrease µ by a factor of 10 each time. Coded in Fortran. Computation easily distributes.

27 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 17 Simulation Results Compare ρesdp as solved by BCGD-barrier with ESDP as solved by Sedumi 1.05 Sturm (with the interface to Sedumi coded by Wang et al).

28 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 17 Simulation Results Compare ρesdp as solved by BCGD-barrier with ESDP as solved by Sedumi 1.05 Sturm (with the interface to Sedumi coded by Wang et al). Anchors and sensors x true 1,..., x true n uniformly distributed in [.5,.5] 2, m =.9n. (i, j) A whenever x true i x true j < rr. Set d ij = x true i x true j (1 + nf ɛ ij ) +, where ɛ ij N(0, 1).

29 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 17 Simulation Results Compare ρesdp as solved by BCGD-barrier with ESDP as solved by Sedumi 1.05 Sturm (with the interface to Sedumi coded by Wang et al). Anchors and sensors x true 1,..., x true n uniformly distributed in [.5,.5] 2, m =.9n. (i, j) A whenever x true i x true j < rr. Set d ij = x true i x true j (1 + nf ɛ ij ) +, where ɛ ij N(0, 1). Sensor i is judged as accurately positioned if tr i [Z found ] < nf.

30 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 18 ρesdp BCGD barrier ESDP Sedumi n m nf rr cpu/m ap /err ap cpu(cpus)/m ap /err ap /574/3.4e-4 189(106)/626/2.2e /520/2.8e-3 170(89)/624/3.1e /517/1.1e-2 128(48)/664/1.5e /1626/1.7e (397)/1689/2.7e /1596/8.5e (503)/1653/1.3e /1602/6.2e (417)/1689/1.2e-2 cpu(sec) times are on a HP DL360 workstation, running Linux 3.5. ESDP is solved by Sedumi; cpus:= run time for Sedumi. Set ρ ij = d 2 ij ((1 2 nf) 2 1). m ap := # accurately positioned sensors. err ap := max i accurate. pos. x i x true i.

31 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION sensors, 100 anchors, rr = 0.06, nf = 0.01, solve ρesdp by BCGD-barrier. x true i (shown as ) and x ρ,δ i (shown as ) are joined by blue line segment; anchors are shown as.

32 ON SDP/ESDP RELAXATION OF SENSOR NETWORK LOCALIZATION 20 Conclusion & Ongoing work SDP and ESDP solns are sensitive to measurement noise. Lack soln accuracy certificate (though the trace test works well enough in simulation). ρesdp has more stable solns. Has soln accuracy certificate (which works well enough in simulation). Needs to estimate the noise level δ to set ρ. Can ρ > δ be relaxed? Approximation bounds? Extension to maxmin dispersion problem. Thanks for coming!..