Spherical Euclidean Distance Embedding of a Graph

Transcription

1 Spherical Euclidean Distance Embedding of a Graph Hou-Duo Qi University of Southampton Presented at Isaac Newton Institute Polynomial Optimization August 9, 2013

2 Spherical Embedding Problem The Problem: Given n points in R m, place them on S r (c, R) the sphere in R r with the center at c and the radius R so that some Euclidean distance properties among the n points are best kept. The most interesting cases are when some or all of the parameters (d, c, R) are unknown.

3 0. Notation: Pre-distance matrix, S n h, and Sn + Pre-distance matrix (dissimilarity matrix): D is symmetric D ii = 0 (zero diagonals) D ij 0 (non-negativities) S n, S+, n and Sh n (Hollow subspace): S n := {all n n symmetric matrices}, Sh n := {X Sn : X ii = 0 i} S+ n := the set of all PSD matrices in S n.

4 1. Squared Euclidean Distance Matrix (EDM) A n n matrix D is a (squared) Euclidean Distance Matrix (EDM) if there exist points p 1,..., p n in R r such that D ij = p i p j 2 i, j. Squared pairwise distances are used. R r is called embedding space and r n 1. The smallest such r is the embedding dimension of D.

5 2. The Cone of EDMs The set of all n n EDMs is a closed convex cone.

6 3. Characterization of EDM: Schoenberg (1935), Young and Householder (1938) Schoenberg in (Ann. Math. 1935), and (independently) Young and Householder in (Psychometrika, 1938). D is EDM D S n h and (JDJ) 0, where J = I ee T /n or (J = I 1 n ). Furthermore, let B = 1 2 JDJ, and B has the following decomposition: B = P P T, with P R n r. Let p i = P (i, :), we have D ij = p i p j 2.

7 3. Characterization of EDM: Schoenberg (1935), Young and Householder (1938) Remarks (R1) The Schoenberg-Young-Householder characterization has two steps: The first step is to versify whether a given matrix is EDM. The second step is the embedding step by computing a spectral decomposition. (R2) It has become a major method for data dimension reduction the classical Multidimensional Scaling (cmds). (R3) The matrix JDJ has zero as its eigenvalue. Therefore, the Slater condition is never satisfied for the constraint: JDJ 0.

8 4. Partial Distances among 50 Sensors

9 5. Algorithm: Isomap Many methods are available Euclidean distance matrix completion (Laurent (1997), Wolkowicz, Anjos et. al from 1999 ) Y. Ye and his co-authors on Semi-Definite Programming (SDP) Relaxations (from 2004 ) Kim et. al on Sparse Full SDP (2009, 2012) Moré and Wu (DGSOL package, Argonne National Laboratory, 1999). Several more packages (e.g., PENNON). Isomap by Tenenbaum, Silva, and Langford (Science 2000). Regard the problem as a network (graph) problem. Length of the edge is the distance (not necessarily accurate) Replace the missing distances by the shortest path distances in the graph. Use the Schoenberg-Young-Householder method to recover the locations of the nodes.

10 4 Points Embedding

11 4 Points Embedding by Isomap

12 Computing Nearest EDM Given a pre-distance matrix D, find a true EDM matrix Y that is the nearest to D: min Y D 2 s.t. Y is EDM rank(jy J) r (embedding dimension constraint) By the Schoenberg-Young-Householder characterization, we have Y is EDM Y Sh n, JY J 0. We have a convex quadratic SDP.

13 4 Points Embedding by EMBED (Q. and Yuan 2012)

14 6. Another Characterization of EDM Hayden and Wells (SIMAX, 1990) and Gaffke and Mathar (Metrika,1989): D is EDM D S n h ( Kn +), where K n + := { A S n : x T Ax 0, x e }. Note: K+ n is a closed convex cone. K+ n as projected spectrahedra: K+ n = { A (A, t 0) such that A tee T 0 } K+ n as set-copositive cone. Conic Formulation of the nearest EDM (Q. and Yuan 2012): min Y D 2 s.t. Y Sh n ( Kn +), and rank(jxj) r.

15 7. Dealing with Spherical Constraints We now want to place n points on a sphere: x i = R. We assume the center to be the (n + 1)th point x n+1 so that x i x n+1 2 = R 2, i = 1,..., 2. The formulation of optimization problems with spherical constraints takes the following form: min / max f(y ) s.t. Y S n h ( Kn+1 + ) rank(jxj) r Y 1(n+1) = Y i(n+1), i = 2,..., n. When there are no rank constraint, the problem is often convex (many such problems from geometric embedding of graphs).

16 8. Smallest Hypersphere Representation of a Grpah Def. Let G = (V, E) be a graph with V = n. A unit-distance representation of g is a system of n vectors (p 1,..., p n ) in a Euclidean space such that Def. If furthermore, p i p j = 1 (i, j) E. p i = p j i, j V the system is called a hypersphere representation of G. Unit-distance realization of Petersen graph on plane

17 8. Smallest Hypersphere Representation of a Graph Finding the smallest radius of a hypersphere representation (Lovász ( 09), Silva and Tuncel ( 10)) It is known t h (G) := min s.t. EDM formulation: min s.t. t diag(x) = te X ii 2X ij + X jj = 1, X S n +, t R. 2t h (G) + 1 ϑ(g) = 1. Y 1(n+1) Y S n h ( Kn+1 + ) Y ij = 1 (i, j) E Y 1(n+1) = Y i(n+1), i = 2,..., n. (i, j) E

18 9. Lovász-theta Function Def. An orthonormal representation of G is a system {p 1,..., p n } of unit vectors in a Euclidean distance space such that p i, p j = 0 (i, j) E. Theorem 5, Lovász ( 79): Let (p 1,..., p n ) range over all orthonormal representations of G and d over all unit vectors. Then ϑ(g) = max n ( d, p i ) 2. i=1 SDP formulation: ϑ(g) = max J, X s.t. I, X = 1 X ij = 0, (i, j) E X 0.

19 From Projection to Euclidean Distance We have d p i 2 = d 2 + p i 2 2 d, p i = 2 2 d, p i. Hence ( d, p i ) 2 = 1 4 ( 1 d pi 2) 2. Under the condition (part of Schrijver s ϑ function): d, p i 0, we have max ( d, p i ) 2 min 1 ( d pi 2) 2 4 This leads to the following EDM problem p(g) := min 1 n ( 4 i=1 d pi 2) 2 s.t. p i = 1; d = 1; p i, p j = 0 (i, j) E.

20 A Quantity that may be interesting For a given graph, define the quantity q(g) such that p(g) + q(g) = n. Let SOL(G) denote the solution set of the SDP of ϑ function. Define τ ϑ := b(ϑ), n where b(ϑ) := max s.t. For vertex-transitive graphs n i=1 Bii B SOL(G). τ ϑ = 1.

21 Bound that measures Distortion Define and r ϑ := n/ϑ(g) t ϑ := r 1 τ (r 1) 2 + 2r(1 τ). Define the distortion constant d ϑ by Claim (Bound of Distortion): d ϑ := t ϑ τ ϑ d 2 ϑϑ(g) q(g) ϑ(g). Remark: d ϑ hard to calculate. But for vertex-transitive graphs, we have d ϑ = 1.

22 Is Triangle Inequality l 2 Metric? One can add triangle inequalities to SDP to strengthen ϑ function: X ik + X jk X ij + X kk (i, j, k V ). Let X 0 admit the Gram representation: Therefore, l 2 -metric: which implies X = P T P. p i p k 2 = X ii + X kk 2X ik. p i p k 2 + p j p k 2 p i p k 2 X ik + X jk X ik (X kk + X jj ).

23 A Wild Guess The close τ ϑ to 1, the less room that adding cut (triangle) inequalities can strengthen ϑ(g). Example is vertex-transitive graphs. We can measure this by computing the ratio: Both are convex problems. ϑ(g) q(g)