Spherical Euclidean Distance Embedding of a Graph Hou-Duo Qi University of Southampton Presented at Isaac Newton Institute Polynomial Optimization August 9, 2013 Spherical Embedding Problem The Problem: Given n points in <m, place them on Sr(c; R) { the sphere in <r with the center at c and the radius R so that some Euclidean distance proper- ties among the n points are \best" kept. The most interesting cases are when some or all of the parame- ters (d; c; R) are unknown. n n 0. Notation: Pre-distance matrix, Sh , and S+ Pre-distance matrix (dissimilarity matrix): I D is symmetric I Dii = 0 (zero diagonals) I Dij ≥ 0 (non-negativities) n n n S , S+, and Sh (Hollow subspace): n I S := fall n × n symmetric matricesg ; n n I Sh := fX 2 S : Xii = 0 8 ig n n I S+ := the set of all PSD matrices in S : 1. Squared Euclidean Distance Matrix (EDM) A n × n matrix D is a (squared) Euclidean Distance r Matrix (EDM) if there exist points p1;:::; pn in < such that 2 Dij = kpi − pjk 8 i; j: ? Squared pairwise distances are used. ? <r is called embedding space and r ≤ n − 1. ? The smallest such r is the embedding dimension of D. 2. The Cone of EDMs I The set of all n × n EDMs is a closed convex cone. 3. Characterization of EDM: Schoenberg (1935), Young and Householder (1938) I Schoenberg in (Ann. Math. 1935), and (independently) Young and Householder in (Psychometrika, 1938). n D is EDM () D 2 Sh and − (JDJ) 0; where 1 J = I − eeT =n or (J = I − ): n I Furthermore, let 1 B = − JDJ; 2 and B has the following decomposition: B = PP T ; with P 2 <n×r: Let pi = P (i; :), we have 2 Dij = kpi − pjk : 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: 4. Partial Distances among 50 Sensors 5. Algorithm: Isomap I Many methods are available I Euclidean distance matrix completion (Laurent (1997), Wolkowicz, Anjos et. al from 1999 {) I Y. Ye and his co-authors on Semi-Definite Programming (SDP) Relaxations (from 2004 {) I Kim et. al on Sparse Full SDP (2009, 2012) I Mor´eand Wu (DGSOL package, Argonne National Laboratory, 1999). I Several more packages (e.g., PENNON). I Isomap by Tenenbaum, Silva, and Langford (Science 2000). I Regard the problem as a network (graph) problem. Length of the edge is the distance (not necessarily accurate) I Replace the missing distances by the shortest path distances in the graph. I Use the Schoenberg-Young-Householder method to recover the locations of the nodes. 4 Points Embedding 4 Points Embedding by Isomap Computing Nearest EDM I Given a pre-distance matrix D, find a true EDM matrix Y that is the nearest to D: min kY − Dk2 s.t. Y is EDM rank(JYJ) ≤ r (embedding dimension constraint) I By the Schoenberg-Young-Householder characterization, we have n Y is EDM () Y 2 Sh ; −JYJ 0: I We have a convex quadratic SDP. 4 Points Embedding by EMBED (Q. and Yuan 2012) 6. Another Characterization of EDM I Hayden and Wells (SIMAX, 1990) and Gaffke and Mathar (Metrika,1989): n n D is EDM () D 2 Sh \ (−K+); where n n n T ?o K+ := A 2 S : x Ax ≥ 0; x 2 e : n Note: K+ is a closed convex cone. n I K+ as projected spectrahedra: n T K+ = A j (A; t ≥ 0) such that A − tee 0 n I K+ as set-copositive cone. I Conic Formulation of the nearest EDM (Q. and Yuan 2012): 2 n n min kY −Dk s.t. Y 2 Sh \(−K+); and rank(JXJ) ≤ r: 7. Dealing with Spherical Constraints I We now want to place n points on a sphere: kxik = R: I We assume the center to be the (n + 1)th point xn+1 so that 2 2 kxi − xn+1k = R ; 8 i = 1;:::; 2: I The formulation of optimization problems with spherical constraints takes the following form: min = max f(Y ) n n+1 s.t. Y 2 Sh \ (−K+ ) rank(JXJ) ≤ r Y1(n+1) = Yi(n+1); i = 2; : : : ; n: I When there are no rank constraint, the problem is often convex (many such problems from geometric embedding of graphs). 8. Smallest Hypersphere Representation of a Grpah I Def. Let G = (V; E) be a graph with jV j = n.A unit-distance representation of g is a system of n vectors (p1;:::; pn) in a Euclidean space such that kpi − pjk = 1 8 (i; j) 2 E: I Def. If furthermore, kpik = kpjk 8 i; j 2 V the system is called a hypersphere representation of G. Unit-distance realization of Petersen graph on plane 8. Smallest Hypersphere Representation of a Graph I Finding the smallest radius of a hypersphere representation (Lov´asz('09), Silva and Tuncel ('10)) th(G) := min t s.t. diag(X) = te Xii − 2Xij + Xjj = 1; 8 (i; j) 2 E n X 2 S+; t 2 <: I It is known 1 2th(G) + = 1: #(G) I EDM formulation: min Y1(n+1) n n+1 s.t. Y 2 Sh \ (−K+ ) Yij = 1 8 (i; j) 2 E Y1(n+1) = Yi(n+1); i = 2; : : : ; n: 9. Lov´asz-thetaFunction I Def. An orthonormal representation of G is a system fp1;:::; png of unit vectors in a Euclidean distance space such that hpi; pji = 0 8 (i; j) 62 E: Theorem 5, Lov´asz('79): Let (p1;:::; pn) range over all orthonormal representations of G and d over all unit vectors. Then n X 2 #(G) = max (hd; pii) : i=1 I SDP formulation: #(G) = max hJ; Xi s.t. hI;Xi = 1 Xij = 0; 8 (i; j) 2 E X 0: From Projection to Euclidean Distance I We have 2 2 2 kd − pik = kdk + kpik − 2hd; pii = 2 − 2hd; pii: Hence 1 2 (hd; p i)2 = 1 − kd − p k2 : i 4 i 0 I Under the condition (part of Schrijver's # function): hd; pii ≥ 0; we have 1 2 max (hd; p i)2 () min kd − p k2 i 4 i I This leads to the following EDM problem 1 Pn 22 p(G) := min 4 i=1 kd − pik s.t. kpik = 1; kdk = 1; hpi; pji = 0 8 (i; j) 2 E: A Quantity that may be interesting I For a given graph, define the quantity q(G) such that p pp(G) + pq(G) = n: I Let SOL(G) denote the solution set of the SDP of # function. Define b(#) τ := p ; # n where Pn p b(#) := max i=1 Bii s.t. B 2 SOL(G): I For vertex-transitive graphs τ# = 1: Bound that measures Distortion I Define p r# := n=#(G) and 1 p t := r − (r − 1)2 + 2r(1 − τ): # τ I Define the distortion constant d# by d# := t#τ# I Claim (Bound of Distortion): 2 d##(G) ≤ q(G) ≤ #(G): I Remark: d# hard to calculate. But for vertex-transitive graphs, we have d# = 1: Is Triangle Inequality `2 Metric? I One can add triangle inequalities to SDP to strengthen # function: Xik + Xjk ≤ Xij + Xkk (i; j; k 2 V ): I Let X 0 admit the Gram representation: X = P T P: Therefore, 2 kpi − pkk = Xii + Xkk − 2Xik: 2 I ` -metric: 2 2 2 kpi − pkk + kpj − pkk ≥ kpi − pkk which implies 1 X + X ≤ X + (X + X ): ik jk ik 2 kk jj A Wild Guess I The close τ# to 1, the less room that adding cut (triangle) inequalities can strengthen #(G). I Example is vertex-transitive graphs. I We can measure this by computing the ratio: #(G) q(G) Both are convex problems..