Spherical Euclidean Distance Embedding of a Graph

Home , Euclidean distance

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 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 := {all n × n symmetric matrices} , n n I Sh := {X ∈ S : Xii = 0 ∀ i} 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 ∀ i, j.

? Squared pairwise distances are used. ?

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 ∈ 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 ∈

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 ∈ 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 ∈ Sh ∩ (−K+), where

n n n T ⊥o K+ := A ∈ S : x Ax ≥ 0, x ∈ e .

n Note: K+ is a closed convex cone. n I K+ as projected spectrahedra: n T K+ = A | (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 ∈ 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 , ∀ 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 ∈ 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 |V | = n.A unit-distance representation of g is a system of n vectors (p1,..., pn) in a Euclidean space such that

kpi − pjk = 1 ∀ (i, j) ∈ E.

I Def. If furthermore,

kpik = kpjk ∀ i, j ∈ 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, ∀ (i, j) ∈ E n X ∈ S+, t ∈ <.

I It is known 1 2th(G) + = 1. ϑ(G) I EDM formulation:

min Y1(n+1) n n+1 s.t. Y ∈ Sh ∩ (−K+ )

Yij = 1 ∀ (i, j) ∈ E

Y1(n+1) = Yi(n+1), i = 2, . . . , n. 9. Lov´asz-thetaFunction

I Def. An orthonormal representation of G is a system {p1,..., pn} of unit vectors in a Euclidean distance space such that hpi, pji = 0 ∀ (i, j) 6∈ 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 ϑ(G) = max (hd, pii) . i=1

I SDP formulation: ϑ(G) = max hJ, Xi s.t. hI,Xi = 1 Xij = 0, ∀ (i, j) ∈ 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 ϑ 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 ∀ (i, j) ∈ E. A Quantity that may be interesting

I For a given graph, deﬁne the quantity q(G) such that √ pp(G) + pq(G) = n.

I Let SOL(G) denote the solution set of the SDP of ϑ function. Deﬁne b(ϑ) τ := √ , ϑ n where Pn √ b(ϑ) := max i=1 Bii s.t. B ∈ SOL(G).

I For vertex-transitive graphs

τϑ = 1. Bound that measures Distortion

I Deﬁne p rϑ := n/ϑ(G) and 1 p t := r − (r − 1)2 + 2r(1 − τ). ϑ τ

I Deﬁne the distortion constant dϑ by

dϑ := tϑτϑ

I Claim (Bound of Distortion):

2 dϑϑ(G) ≤ q(G) ≤ ϑ(G).

I Remark: dϑ hard to calculate. But for vertex-transitive graphs, we have dϑ = 1. Is Triangle Inequality `2 Metric?

I One can add triangle inequalities to SDP to strengthen ϑ function:

Xik + Xjk ≤ Xij + Xkk (i, j, k ∈ 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 τϑ to 1, the less room that adding cut (triangle) inequalities can strengthen ϑ(G).

I Example is vertex-transitive graphs.

I We can measure this by computing the ratio:

ϑ(G) q(G)

Both are convex problems.