Facial reduction for Euclidean distance matrix problems

Transcription

1 Facial reduction for Euclidean distance matrix problems Nathan Krislock Department of Mathematical Sciences Northern Illinois University, USA Distance Geometry: Theory and Applications DIMACS Center, Rutgers University, USA July 26, 2016 Joint work with Dmitriy Drusvyatskiy (University of Washington), Yuen-Lam Voronin (University of Colorado), and Henry Wolkowicz (University of Waterloo) Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

2 Outline 1 Euclidean Distance Matrices 2 Facial Reduction 3 Facial Reduction for EDM Completion 4 Noisy EDM Completion Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

3 Outline 1 Euclidean Distance Matrices 2 Facial Reduction 3 Facial Reduction for EDM Completion 4 Noisy EDM Completion Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

4 Sensor Network Localization n = 100, m = 9, R = 2 Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

5 Protein Structure Determination Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

6 Euclidean Distance Matrices Euclidean distance matrices (EDMs) An n n matrix D is an EDM if The embedding dimension of D: E n = the set of all n n EDMs EDM completion p 1,..., p n R k : D ij = p i p j 2 2 (1) dim(d) = min{k : (1) holds} find D E n such that D ij = D ij, ij E dim(d) k Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

7 Euclidean Distance Matrices EDMs and semidefinite matrices Let p 1,..., p n R k and D be their EDM Let Y S n be the Gram matrix: Y ij = p i, p j, ij Then Y is a semidefinite matrix: Y S n + Therefore: K(S n +) = E n D ij = p i p j 2 2 = p i, p i 2 p i, p j + p j, p j = Y ii 2Y ij + Y jj =: K(Y ) ij Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

8 Euclidean Distance Matrices Translation invariant: D = D but Y Y D = K(Y ) D = K(Y ) Centered matrices: S n c := {Y S n : Ye = 0}, e = [ 1 1 ] T K : S n + S n c E n is a linear bijection whose inverse is given by K (D) = 1 2 J offdiag(d)j, J = I 1 n eet If D E n, then dim(d) = rank(k (D)) Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

9 Euclidean Distance Matrices EDM completion find Y S+ n Sc n such that K(Y ) ij = D ij, ij E rank(y ) k non-convex and NP-hard Semidefinite relaxation find Y S+ n Sc n such that K(Y ) ij = D ij, ij E convex and solvable in polynomial-time by an interior-point method only problems of small size can be solved efficiently highly degenerate: not strictly feasible facial reduction a much smaller equivalent problem Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

10 Outline 1 Euclidean Distance Matrices 2 Facial Reduction 3 Facial Reduction for EDM Completion 4 Noisy EDM Completion Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

11 Facial Reduction Face of a convex cone K We say a convex cone F K is a face of K (denoted F K) if x, y F whenever x, y K and 1 (x + y) F 2 The minimal face of K containing S K is face(s) := F S F K We say that F K is exposed if there exists φ such that F = K {φ} Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

12 Facial Reduction A simple example minimize 2x 1 + 6x 2 x 3 2x 4 subject to x 1 + x 2 + x 3 + x 4 = 1 x 1 x 2 x 3 = -1 x 1, x 2, x 3, x 4 0 Summing the constraints: 2x 1 + x 4 = 0 x 1 = x 4 = 0 Restrict problem to the face { x R 4 + : x 1 = x 4 = 0 } minimize 6x 2 x 3 subject to x 2 + x 3 = 1 x 2, x 3 0 Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

13 Facial Reduction Linear programming (LP) minimize subject to c T x Ax = b x R n + Certificate of non-strict feasibility Suppose y R m such that 0 A T y 0 and b T y = 0 If Ax = b and x 0, then y T Ax = y T b = (A T y) T x = 0 = x i = 0, i supp(a T y) Therefore, the feasible set of (LP) is contained in the face { } R n + {A T y} = x R n + : x i = 0, i supp(a T y) Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

14 Facial Reduction Semidefinite programming (SDP) minimize C, X subject to AX = b X S+ n Certificate of non-strict feasibility Suppose y R m such that 0 A y 0 and b T y = 0 If AX = b and X 0, then y T AX = y T b = A y, X = 0 = X (A y) = (A y)x = 0 Therefore, the feasible set of (SDP) is contained in the face S+ n {A y} = { X S+ n : range(x ) null(a y) } Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

15 Facial Reduction Faces of the semidefinite cone Let Q = [ U V ] R n n be orthogonal and F := { X S+ n : range(x ) range(u) } S+. n Then F = US k +U T. Semidefinite facial reduction [Borwein & Wolkowicz (1981)] { X S n + : A i, X = b i, i } US k +U T Then substituting X = UZU T, (SDP) is equivalent to: minimize C, UZU T subject to A i, UZU T = b i, i Z S+ k Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

16 Facial Reduction Faces of the semidefinite cone Let Q = [ U V ] R n n be orthogonal and F := { X S+ n : range(x ) range(u) } S+. n Then F = US k +U T. Semidefinite facial reduction [Borwein & Wolkowicz (1981)] { X S n + : A i, X = b i, i } US k +U T Then substituting X = UZU T, (SDP) is equivalent to: minimize U T CU, Z subject to U T A i U, Z = b i, i Z S+ k Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

17 Outline 1 Euclidean Distance Matrices 2 Facial Reduction 3 Facial Reduction for EDM Completion 4 Noisy EDM Completion Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

18 Facial Reduction for EDM Completion Theorem: Facial reduction for EDM [K. & Wolkowicz (2010)] Let D E p with dim(d) = k, F := { Y S n + S n c : K(Y ) ij = D ij, i, j = 1,..., p }, the columns of Ū Rp (k+1) form a basis for range ([ K (D) [Ū ] 0 U :=. 0 I n p e ]), and Then face(f) = ( US n p+k+1 + U T ) S n c. Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

19 Facial Reduction for EDM Completion Theorem: Constraint reduction [K. & Wolkowicz (2010)] Let D E p with dim(d) = k, F := { Y S n + S n c : K(Y ) ij = D ij, i, j = 1,..., p }, and U be defined as in the previous theorem. Let β {1,..., p} such that dim(d[β, β]) = k. Then F = { ( ) } Y US n p+k+1 + U T Sc n : K(Y ) ij = D ij, i, j β. Example : n = 1000, p = 100, β = 3 variables : , constraints : Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

20 Facial Reduction for EDM Completion Algorithm for computing the face F := { Y S n + S n c : K(Y ) ij = D ij, ij E } Find some cliques α i in the graph ( ) Let F i := U i S n α i +t i +1 + Ui T Sc n be the corresponding faces Compute U R n (t+1) full column rank such that range(u) = i range(u i ) Then: F ( US t+1 + UT ) S n c Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

21 Facial Reduction for EDM Completion Protein structure determination Protein 2L30 from the PDB 393 cliques: 269 2D D SDP size: equality constraints: Babak Alipanahi, Nathan Krislock, Ali Ghodsi, Henry Wolkowicz, Logan Donaldson, and Ming Li. (2012) Determining protein structures from NOESY distance constraints by semidefinite programming. Journal of Computational Biology. Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

22 Facial Reduction for EDM Completion Theorem: EDM completion [K. & Wolkowicz (2010)] Let D be a partial EDM and F := { Y S n + S n c : K(Y ) ij = D ij, ij E } F ( US t+1 + UT ) S n c = (UV )S t +(UV ) T If Ȳ F and β is a clique with dim(d[β, 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]) 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 (NIU) Facial reduction for EDM problems DGTA / 29

23 Facial Reduction for EDM Completion Face representation approach n R Time RMSD (%R) s 2e s 7e m 53 s 5e m 46 s 8e-7 Point representation approach n R Time RMSD (%R) s 1e s 3e m 27 s 6e m 55 s 9e-12 Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

24 Outline 1 Euclidean Distance Matrices 2 Facial Reduction 3 Facial Reduction for EDM Completion 4 Noisy EDM Completion Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

25 Noisy EDM Completion Multiplicative noise model d ij = p i p j 2 (1 + σε ij ), for all ij E ε ij is normally distributed with mean 0 and standard deviation 1 σ 0 is the noise factor D ij = d 2 ij Point representation approach (no refinement used) n σ R Time RMSD (%R) e s e s e s 500 Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

26 Noisy EDM Completion Exposing vector representation of faces Recall we had faces in primal form: ( F i = U i S n α i +t i +1 + Ui T ) S n c The exposing vector Φ i for the face F i gives a dual representation: F i = S n + {Φ i } The intersection of the faces can be easily computed by F := i { } F i = S+ n Φ i i Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

27 Noisy EDM Completion Exposing Vector Algorithm For each clique α Y α := nearest pos. semidef. rank-r matrix to K (D[α, α]) Φ α := exposing vector of face({y α }) extended to S n by adding zeros Φ := nearest pos. semidef. rank-(n r 1) matrix to α Φ α Let U R n r, such that U T e = 0, be eigenvectors of Φ for the smallest r eigenvalues { } Solve min ij E K(UZUT ) ij D ij 2 : Z S+ r Return P = UZ 1/2 Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

28 Noisy EDM Completion Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

29 Noisy EDM Completion Exposing Vector Algorithm + Refinement (10% anchors) Specifications Time RMSD (%R) n σ R initial refine initial refine s 0.1 s s 2 s s 2 s s 2 s s 0.3 s s 6 s s 6 s s 6 s s 14 s s 20 s s 44 s Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29

30 Summary EDM problems are highly degenerate due to non-strict-feasibility Facial reduction is a powerful technique for taking advantage of this degeneracy The exposing vector approach is a very effective approach for noisy EDM problems Nathan Krislock (NIU) Facial reduction for EDM problems DGTA / 29