Semidefinite Facial Reduction for Euclidean Distance Matrix Completion

Transcription

1 Semidefinite Facial Reduction for Euclidean Distance Matrix Completion Nathan Krislock, Henry Wolkowicz Department of Combinatorics & Optimization University of Waterloo First Alpen-Adria Workshop on Optimization University of Klagenfurt, Austria June 3-6, 2010 Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

2 Sensor Network Localization n = 100, m = 9, R = 2 Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

3 Molecular Conformation Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

4 Previous Work I Abdo Y. Alfakih, Amir Khandani, and Henry Wolkowicz. Solving Euclidean distance matrix completion problems via semidefinite programming. Comput. Optim. Appl., 12(1-3):13 30, L. Doherty, K. S. J. Pister, and L. El Ghaoui. Convex position estimation in wireless sensor networks. In INFOCOM Twentieth Annual Joint Conference of the IEEE Computer and Communications Societies. Proceedings. IEEE, volume 3, pages vol.3, Pratik Biswas and Yinyu Ye. Semidefinite programming for ad hoc wireless sensor network localization. In IPSN 04: Proceedings of the 3rd international symposium on Information processing in sensor networks, pages 46 54, New York, NY, USA, ACM. Pratik Biswas, Tzu-Chen Liang, Kim-Chuan Toh, Ta-Chung Wang, and Yinyu Ye. Semidefinite programming approaches for sensor network localization with noisy distance measurements. IEEE Transactions on Automation Science and Engineering, 3: , Pratik Biswas, Kim-Chuan Toh, and Yinyu Ye. A distributed SDP approach for large-scale noisy anchor-free graph realization with applications to molecular conformation. SIAM Journal on Scientific Computing, 30(3): , Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

5 Previous Work II Zizhuo Wang, Song Zheng, Yinyu Ye, and Stephen Boyd. Further relaxations of the semidefinite programming approach to sensor network localization. SIAM Journal on Optimization, 19(2): , Sunyoung Kim, Masakazu Kojima, and Hayato Waki. Exploiting sparsity in SDP relaxation for sensor network localization. SIAM Journal on Optimization, 20(1): , Ting Pong and Paul Tseng. (Robust) Edge-based semidefinite programming relaxation of sensor network localization. Mathematical Programming, Published online. Nathan Krislock and Henry Wolkowicz. Explicit sensor network localization using semidefinite representations and facial reductions. To appear in SIAM Journal on Optimization, Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

6 Summary We developed a theory of semidefinite facial reduction for the EDM completion problem Using this theory, we can transform the SDP relaxations into much smaller equivalent problems We developed a highly efficient algorithm for EDM completion: SDP solver not required can solve problems with up to 100,000 sensors in a few minutes on a laptop computer obtained very high accuracy for noiseless problems our running times are highly competitive with other SDP-based codes Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

7 Outline 1 Euclidean Distance Matrices Introduction Euclidean Distance Matrices and Semidefinite Matrices Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

8 Outline 1 Euclidean Distance Matrices Introduction Euclidean Distance Matrices and Semidefinite Matrices 2 Facial Reduction Semidefinite Facial Reduction Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

9 Outline 1 Euclidean Distance Matrices Introduction Euclidean Distance Matrices and Semidefinite Matrices 2 Facial Reduction Semidefinite Facial Reduction 3 Results Facial Reduction Theory for EDM Completion Numerical Results Noisy Problems Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

10 Outline 1 Euclidean Distance Matrices Introduction Euclidean Distance Matrices and Semidefinite Matrices 2 Facial Reduction Semidefinite Facial Reduction 3 Results Facial Reduction Theory for EDM Completion Numerical Results Noisy Problems Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

11 Introduction Euclidean Distance Matrices An n n matrix D is an EDM if p 1,..., p n R r D ij = p i p j 2, for all i, j = 1,..., n E n := set of all n n EDMs Embedding Dimension of D E n { } embdim(d) := min r : p 1,..., p n R r s.t. D ij = p i p j 2, for all i, j Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

12 Introduction Partial Euclidean Distance Matrices D is a partial EDM in R r if every entry D ij is either specified or unspecified, diag(d) = 0 for all α {1,..., n}, if the principal submatrix D[α] is fully specified, then D[α] is an EDM with embdim(d[α]) r Problem: determine if there is a completion ˆD E n with embdim( ˆD) = r Graph of a Partial EDM G = (N, E, ω) weighted graph with nodes N := {1,..., n} edges E := { ij : i j, and D ij is specified } weights ω R E + with ω ij := D ij Problem: determine if there is a realization p : N R r of G Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

13 Introduction n = 100, m = 9, R = 2 Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

14 Outline 1 Euclidean Distance Matrices Introduction Euclidean Distance Matrices and Semidefinite Matrices 2 Facial Reduction Semidefinite Facial Reduction 3 Results Facial Reduction Theory for EDM Completion Numerical Results Noisy Problems Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

15 EDMs and Semidefinite Matrices Linear Transformation K Let D E n be given by the points p 1,..., p n R r (or P R n r ) Let Y := PP T = (pi T p j ) S+ n D ij = p i p j 2 = pi T p i + pj T p j 2pi T p j = Y ii + Y jj 2Y ij Thus D = K(Y ), where: K(Y ) := diag(y )e T + ediag(y ) T 2Y K(S+) n = E n (but not one-to-one) Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

16 EDMs and Semidefinite Matrices Moore-Penrose Pseudoinverse of K K (D) = 1 2 J [offdiag(d)] J where J := I 1 n eet Theorem: (Schoenberg, 1935) A matrix D with diag(d) = 0 is a Euclidean distance matrix if and only if K (D) is positive semidefinite. Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

17 EDMs and Semidefinite Matrices Properties of K and K S n C := {Y Sn : Ye = 0} and S n H := {D Sn : diag(d) = 0} K (SC n ) = Sn H and K (SH n ) = Sn C K ( S+ n SC n ) = E n and K (E n ) = S+ n SC n ( ) embdim(d) = rank K (D), for all D E n Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

18 EDMs and Semidefinite Matrices Vector Formulation Find p 1,..., p n R r such that p i p j 2 = D ij, ij E Matrix Formulation using K Find P R n r such that H K(Y ) = H D, Y = PP T Semidefinite Programming (SDP) Relaxation Find Y S+ n SC n such that H K(Y ) = H D Vector/Matrix Formulation is non-convex and NP-hard SDP Relaxation is tractable, but only problems of limited size can be directly handled by an SDP solver To solve this SDP, we use facial reduction to obtain a much smaller equivalent problem Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

19 Outline 1 Euclidean Distance Matrices Introduction Euclidean Distance Matrices and Semidefinite Matrices 2 Facial Reduction Semidefinite Facial Reduction 3 Results Facial Reduction Theory for EDM Completion Numerical Results Noisy Problems Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

20 Semidefinite Facial Reduction An LP Example minimize 2x 1 + 6x 2 x 3 2x 4 + 7x 5 subject to x 1 + x 2 + x 3 + x 4 = 1 x 1 x 2 x 3 + x 5 = -1 x 1, x 2, x 3, x 4, x 5 0 Summing the constraints: 2x 1 + x 4 + x 5 = 0 x 1 = x 4 = x 5 = 0 Restrict LP to the face { x R 5 + : x 1 = x 4 = x 5 = 0 } R 5 + minimize 6x 2 x 3 subject to x 2 + x 3 = 1 x 2, x 3 0 Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

21 Semidefinite Facial Reduction The Minimal Face If K E is a convex cone and S K, then face(s) := F Proposition: S F K Let K E be a convex cone and let S F K. If S and convex, then: face(s) = F if and only if S relint(f). Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

22 Semidefinite Facial Reduction Representing Faces of S n + If F S n + and X relint(f) with rank(x) = t, then F = US t +U T and relint(f) = US t ++U T, where U R n t has full column rank and range(u) = range(x). Semidefinite Facial Reduction If: Then: face ({ X S n + : A i, X = b i, i }) = US t +U T minimize C, X subject to A i, X = b i, i X S+ n = minimize C, UZU T subject to A i, UZU T = b i, i Z S+ t Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

23 Outline 1 Euclidean Distance Matrices Introduction Euclidean Distance Matrices and Semidefinite Matrices 2 Facial Reduction Semidefinite Facial Reduction 3 Results Facial Reduction Theory for EDM Completion Numerical Results Noisy Problems Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

24 Facial Reduction Theory for EDM Completion Theorem: Clique Facial Reduction Let: Then: D E k t := embdim( D) F := { Y S k + : K(Y ) = D } where Ū := [ U C e ] face( F) = ŪSt+1 + ŪT U C R k t full column rank and range(u C ) = range(k ( D)) Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

25 Facial Reduction Theory for EDM Completion Theorem: Extending the Facial Reduction Let D be a partial EDM, α {1,..., n}, and: If: Then: F := { Y S+ n SC n : H K(Y ) = H D} F α := { Y S+ n SC n : H[α] K(Y [α]) = H[α] D[α]} { } F α := Y S α + : H[α] K(Y ) = H[α] D[α] where U := face(f α ) face( F α ) ŪSt+1 + ŪT [Ū ] 0 R 0 I n (n α +t+1) ( US n α +t+1 + U T ) S n C Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

26 Facial Reduction Theory for EDM Completion Theorem: Distance Constraint Reduction Let D be a partial EDM, α {1,..., n}, and: F α := { Y S+ n SC n : H[α] K(Y [α]) = H[α] D[α]} { } F α := Y S α + : H[α] K(Y ) = H[α] D[α] face( F α ) ŪSt+1 + ŪT If Ȳ F α and β α is a clique with embdim(d[β]) = t, then: F α = where U := { Y ( US n α +t+1 + U T ) S n C : K(Y [β]) = D[β] } [Ū ] 0 R 0 I n (n α +t+1) Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

27 Facial Reduction Theory for EDM Completion Theorem: Disjoint Subproblems Let F be as above. Let {α i } l i=1 be disjoint subsets, α := α i, and : F i := { Y S+ n SC n : H[α i] K(Y [α i ]) = H[α i ] D[α i ] } { } F i := Y S α i + : H[α i] K(Y ) = H[α i ] D[α i ] If face( F i ) ŪiS t i +1 + ŪT i, then face(f) ( US n α +t+1 + U T ) S n C Ū where U := Ū l 0 Rn (n α +t+1) 0 0 I Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

28 Facial Reduction Theory for EDM Completion Facial Reduction Algorithm For each node i = 1,..., n, find a clique C i containing i Let C n+1 be the clique of anchors ( ) Let F i := U i S n C i +t i +1 + Ui T SC n be the corresponding faces Compute U R n (t+1) full column rank such that Then: range(u) = n+1 i=1 range(u i ) face ({ Y S n + S n C : H K(Y ) = H D}) ( US t+1 + UT ) S n C Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

29 Facial Reduction Theory for EDM Completion Rigid Intersection C i C j Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

30 Facial Reduction Theory for EDM Completion Lemma: Rigid Face Intersection Suppose: and U Then: 1, U Satisfies: U 1 0 I 0 U 1 = U 1 0 and U 2 = 0 U 2 0 I 0 U 2 2 Rk (t+1) full column rank with range(u 1 ) = range(u U 1 U := U 1 or U := U 2 (U 2 ) U 1 U 1 (U 1 ) U 2 U 2 U 2 range(u) = range(u 1 ) range(u 2 ) 2 ) Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

31 Facial Reduction Theory for EDM Completion Clique Union Node Absorption C i Rigid C i C j j C i Non-rigid Ci C j j Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

32 Facial Reduction Theory for EDM Completion Theorem: Euclidean Distance Matrix Completion Let D be a partial EDM and : If Ȳ F := { Y S+ n SC n : H K(Y ) = H D} ( ) face(f) US+ t+1 UT SC n = (UV )St +(UV ) T F and β is a clique with embdim(d[β]) = t, then: Ȳ = (UV ) Z (UV ) T, where Z is the unique solution of (JU[β, :]V )Z (JU[β, :]V ) T = K (D[β]) (1) F = { Ȳ } D := K( P P T ) E n is the unique completion of D, where P := UV Z 1/2 R n t Note: In this case, an SDP solver is not required. Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

33 Outline 1 Euclidean Distance Matrices Introduction Euclidean Distance Matrices and Semidefinite Matrices 2 Facial Reduction Semidefinite Facial Reduction 3 Results Facial Reduction Theory for EDM Completion Numerical Results Noisy Problems Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

34 Numerical Results Random noiseless problems Dimension r = 2 Square region: [0, 1] [0, 1] m = 4 anchors R = radio range Using only Rigid Clique Union and Rigid Node Absorption Error measure: Root Mean Square Deviation ( 1 RMSD := # positioned i positioned ) 1 2 p i pi true 2. Results averaged over 10 instances Used MATLAB on a 2.16 GHz Intel Core 2 Duo with 2 GB of RAM Source code SNLSDPclique available on author s website, released under a GNU General Public License Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

35 Numerical Results Face Representation Approach # # Sensors CPU sensors R Positioned Time RMSD s 2e s 3e s 2e s 4e s 8e s 7e s 1e s 3e s 7e s 1e s 2e s 1e s 2e m 53 s 7e m 46 s 9e-11 Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

36 Numerical Results Point Representation Approach # # Sensors CPU sensors R Positioned Time RMSD s 5e s 6e s 7e s 7e s 5e s 5e s 8e s 6e s 9e s 7e s 6e s 1e s 8e m 27 s 9e m 55 s 1e-15 Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

37 Outline 1 Euclidean Distance Matrices Introduction Euclidean Distance Matrices and Semidefinite Matrices 2 Facial Reduction Semidefinite Facial Reduction 3 Results Facial Reduction Theory for EDM Completion Numerical Results Noisy Problems Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

38 Noisy Problems Multiplicative Noise Model d 2 ij = p i p j 2 (1 + σε ij ) 2, for all ij E ε ij is normally distributed with mean 0 and standard deviation 1 σ 0 is the noise factor Least Squares Problem minimize ij E v 2 ij subject to p i p j 2 (1 + v ij ) 2 = d 2 n i=1 p i = 0 p 1,..., p n R r ij, for all ij E Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

39 Noisy Problems Point Representation Approach # CPU σ sensors R Time RMSD s 5e-16 1e s 1e-06 1e s 1e-04 1e s 7e s 5e-16 1e s 1e-06 1e s 1e-04 1e s 2e s 1e-15 1e s 1e-06 1e s 1e-04 1e s 2e-01 Note: No refinement technique is used in our numerical tests Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

40 Noisy Problems 1LFB nf = 0.01%, RMSD = LFB nf = 0.1%, RMSD = Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

41 Noisy Problems 1LFB nf = 1%, RMSD = LFB nf = 10%, RMSD = Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41

42 Summary We developed a theory of semidefinite facial reduction for the EDM completion problem Using this theory, we can transform the SDP relaxations into much smaller equivalent problems We developed a highly efficient algorithm for EDM completion: SDP solver not required can solve problems with up to 100,000 sensors in a few minutes on a laptop computer obtained very high accuracy for noiseless problems our running times are highly competitive with other SDP-based codes Nathan Krislock (University of Waterloo) SDP Facial Reduction for EDM Completion WO / 41