A least squares approach for the Discretizable Distance Geometry - PowerPoint PPT Presentation

A least squares approach for the Discretizable Distance Geometry Problem with inexact distances Douglas S. Gon¸ calves Department of Mathematics Universidade Federal de Santa Catarina Distance Geometry Theory and Applications DIMACS - New Jersey - July, 2016 Partially supported by CNPq. Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 1 / 28

Distance Geometry problem Definition (DGP) Given a simple weighted undirected graph G ( V, E, d ) , d : E → R + , and a positive integer K , is there a map x : V → R K such that the constraints � x i − x j � 2 = d 2 ij , ∀{ i, j } ∈ E are satisfied ? Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 2 / 28

Discretizable Distance Geometry problem Definition(DDGP) A DGP is said discretizable if there exists a vertex order { v 1 , v 2 , . . . , v N } ensuring that: (a) G [ { v 1 , v 2 , . . . , v K } ] is a clique; (b) For each i > K : i) { v j , v i } ∈ E , for j = i − K, . . . , i − 2 , i − 1 , ii) V 2 (∆( { v i − K , . . . , v i − 1 } )) > 0 . * The definition ensures that the underlying graph is a chain of ( K + 1) -cliques. Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 3 / 28

Exact distances: a branch-and-prune approach By DDGP assumptions we have that coordinates x i for each vertex v i are obtained by intersecting K spheres: � x i − 1 − x i � 2 d 2 = i − 1 ,i � x i − 2 − x i � 2 d 2 = i − 2 ,i . . . � x i − K − x i � 2 d 2 = i − K,i which leads to at most 2 candidate positions (branching) . Pruning: Direct Distance Feasibility(DDF) |� x h − x i � − d hi | < ǫ, ∀ h : { h, i } ∈ E and h < i − K (Lavor et al., Comp. Optim. App., 52 , 2012) Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 4 / 28

Exact distances: search tree d 14 d 13 d 15 (Liberti et al., Discrete App. Math., 165 , 2014) Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 5 / 28

Exact distances: symmetries and other properties Search space has the structure of a binary tree (with 2 N − K leaf nodes) If pruning distances appear frequently enough it is possible to efficiently explore the search space The number of solutions is a power of 2 Due to the symmetries in the DDGP search tree, it suffices to find the 1st solution: the others can be constructed by partial reflections (Liberti et al., SIAM Review, 56 , 2014) Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 6 / 28

DDGP with noisy distances Consider that exact distances d 2 ij are disturbed by a small noise δ ij d 2 ˜ ij = d 2 ij + δ ij , with | δ ij | ≤ δ , such that � δ d � ≤ √ m δ. Problem: find approximate solutions of � x i − x j � 2 − ˜ d 2 ij = 0 , ∀{ i, j } ∈ E Aim: extend the BP approach for DDGP with noisy distances Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 7 / 28

Noisy distances d 14 d 13 d 15 Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 8 / 28

Noisy distances d 14 d 13 d 15 Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 9 / 28

Noisy distances d 14 d 13 d 15 Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 10 / 28

Noisy distances d 14 d 13 d 15 Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 11 / 28

Noisy distances d 14 d 13 d 15 ¯ d 15 Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 12 / 28

Least-squares, SVD and candidate positions Theorem (Low rank approximation) If σ 1 ≥ σ 2 ≥ · · · ≥ σ r are the nonzero singular values of A ∈ R n × n and A = U Σ V ⊤ , then for each K < r , the distance from A to the closest matrix of rank K is σ K +1 = rank ( B )= K � A − B � 2 , min achieved at B = � K i =1 σ i u i v ⊤ i . Corollary: n � σ 2 rank ( B )= K � A − B � 2 i = min F . i = K +1 (Golub and Van Loan, Matrix Computations , 1996) Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 13 / 28

Candidate positions: 1st candidate ˜ D i : reduced(complete) distance matrix related to { v i − K , . . . , v i − 1 , v i } X i ∈ R K × ( K +1) , X i = [ x i − K . . . x i − 1 x i ] H = I n − 1 G i = − 1 ˜ 2 H ˜ nee ⊤ : centering matrix, D i H : Gram matrix If ˜ G i = U ˜ Σ U ⊤ , then K � ¯ rank ( G )= K � G − ˜ σ k u k u ⊤ G i = arg min G i � 2 = ˜ k , k =1 and, since ¯ G i = ¯ i ¯ X ⊤ X i , candidate positions are given by: X i = (˜ ¯ Σ(1 : K, 1 : K )) 1 / 2 ( U (: , 1 : K )) ⊤ (Sit et al., Bull. Math. Bio., 71 , 2009) Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 14 / 28

Orthogonal Procrustes The first K vectors X = [¯ x i − K . . . x i − 1 ] are used to transform the coordinates ¯ of ¯ x i back to the original reference system: Y = [ x i − K . . . x i − 1 ] (already placed) After centering X c = X ( I − 1 nee ⊤ ) , Y c = Y ( I − 1 nee ⊤ ) , find Q such that Q ⊤ Q = I � QX c − Y c � 2 min F . Given Y c X ⊤ c = U Σ V ⊤ , we have Q = UV ⊤ x ′ i ← Q ¯ x i + t , where t = 1 nY e − Q 1 nXe = y c − Qx c . (Dokmanic et al., IEEE Signal Proces., 32 , 2015) Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 15 / 28

Reflection: 2nd candidate From the assumptions of DDGP, the set { x i − K , . . . , x i − 1 } is affinely independent, generating an affine subspace A of dimension K − 1 . A ⊥ = span { u } Let u be a unit vector orthogonal to A . Then the points in A satisfy u ⊤ x = β A u ⊤ x = β, and the reflection of x i through that u ⊤ x = 0 hyperplane is given by x ′′ i = ( I − 2 uu ⊤ ) x i + 2 βu Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 16 / 28

Consistency Let D i , ˜ D i and ¯ D i be the true, disturbed and approximated reduced distance matrices, respectively, and G i , ˜ G i and ¯ G i their associated Gram matrix. As � G i � 2 = 1 D i � 2 = 1 2 � E i � 2 ≤ 1 2 � E i � F ≤ 1 n ( n − 1) � G i − ˜ 2 � D i − ˜ δ, 2 2 we have that � G i � 2 ≤ 1 n ( n − 1) σ K +1 = � ¯ G i − ˜ G i � 2 ≤ � G i − ˜ ˜ δ. 2 2 σ K +1 → 0 as δ → 0 , implying � ¯ G i − ˜ Therefore ˜ G i � → 0 . But when δ → 0 , ˜ G i → G i , thus ¯ G i → G i . Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 17 / 28

Pruning devices: DDF criterion Direct Distance Feasibility: for all j < i − K : { j, i } ∈ E � � � � x i − x j � 2 − ˜ d 2 � � � ≤ ε 1 . ij How to choose ε 1 ? Let ˜ d be the vector with components ˜ d 2 ij . Choose ε 1 such that MDE ( x ( ε 1 ); ˜ d ) ≤ τ � δ d � , where τ ≥ 1 , x ( ε 1 ) is the first solution found by BP and 1 |� x i − x j � − d ij | � MDE ( x ; d ) = . | E | d ij { i,j }∈ E Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 18 / 28

Rigidity and noisy distances Let x ∈ R KN be a realization of G ( V, E ) , R ∈ R | E |× KN be the rigidity matrix of ( G, x ) and ˜ x the solution of 1 � � 2 � � x i − x j � 2 − ˜ d 2 min . ij 2 x { i,j }∈ E x − x and δ d the vector with entries δ ij = ˜ d 2 ij − d 2 Define δ x = ˜ ij . From the first order Taylor approximation, we have Rδ x = 1 2 δ d . Thus δ x = 1 2 R † δ d . and � δ x � = 1 1 2 � R † �� δ d � = � δ d � . 2 σ r (Anderson et al., SIAM J. Discrete Math., 24 , 2010) Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 19 / 28

Pruning devices: a relaxed DDF criterion Thus, for the solution ˜ x of the perturbed NLSP, we have � d 2 � x j � 2 − ˜ � � � � � 2( x i − x j ) ⊤ ( δx i − δx j ) − δ ij � � ˜ x i − ˜ ≈ � � ≤ 2 � x i − x j �� δx i − δx j � + | δ ij | d ij ) � δ d � ≤ 2 (max d ij ) 2 � δ x � + δ ≤ 2 (max + δ σ r ij ij √ m � � ≤ 2(max d ij ) + 1 δ. σ r ij Therefore, we demand that the approximate solution ¯ x satisfies: ≈ ε 1 � �� � � � � d 2 � d ij ) √ m c 1 + 1 x j � 2 − ˜ ˜ � � � � ¯ x i − ¯ � ≤ γ 2(max δ, ij where γ > 1 and c 1 is an estimate for 1 /σ r . Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 20 / 28

Pruning devices: Singular value ratio Let ˆ D i be the matrix of square distances related to v i and its predecessors(neighbors v j of v i such that j < i ). Missing entries of ˆ D i are obtained from already computed positions x j , j < i . Let G i = − 1 ˆ 2 H ˆ D i H = U Σ V ⊤ A wrong choice of previous candidate positions may forbids the distances in ˆ D i to lead to a realization in R K . Thus, we consider the ratio � K k =1 ˆ σ k ρ = , � n k =1 ˆ σ k and the current tree path is pruned whenever: (1 − ρ ) > ε 2 . Douglas S. Gon¸ calves (UFSC) DDGP - Least squares DGTA 21 / 28