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
S r (c, R) – the sphere in <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.
0. Notation: Pre-distance matrix, Shn , and S+n
Pre-distance matrix (dissimilarity matrix):
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D is symmetric
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Dii = 0 (zero diagonals)
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Dij ≥ 0 (non-negativities)
n , and S n (Hollow subspace):
S n , S+
h
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S n := {all n × n symmetric matrices} ,
I
Shn := {X ∈ S n : Xii = 0 ∀ i}
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n := the set of all PSD matrices in S n .
S+
1. Squared Euclidean Distance Matrix (EDM)
A n × n matrix D is a (squared) Euclidean Distance
Matrix (EDM) if there exist points p1 , . . . , pn in <r
such that
Dij = kpi − pj k2
∀ 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
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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).
D is EDM
⇐⇒
D ∈ Shn and − (JDJ) 0,
where
J = I − eeT /n
I
or
(J = I −
1
).
n
Furthermore, let
1
B = − JDJ,
2
and B has the following decomposition:
B = P P T , with P ∈ <n×r .
Let pi = P (i, :), we have
Dij = kpi − pj k2 .
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
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Many methods are available
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I
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I
I
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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).
I
I
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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.
4 Points Embedding
4 Points Embedding by Isomap
Computing Nearest EDM
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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(JY J) ≤ r
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By the Schoenberg-Young-Householder characterization, we
have
Y
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(embedding dimension constraint)
is EDM
⇐⇒
Y ∈ Shn , −JY J 0.
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):
D is EDM
⇐⇒
n
D ∈ Shn ∩ (−K+
),
where
n
o
n
K+
:= A ∈ S n : xT Ax ≥ 0, x ∈ e⊥ .
I
I
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n is a closed convex cone.
Note: K+
n as projected spectrahedra:
K+
n
K+
= A | (A, t ≥ 0) such that A − teeT 0
n as set-copositive cone.
K+
Conic Formulation of the nearest EDM (Q. and Yuan 2012):
min kY −Dk2
s.t.
n
Y ∈ Shn ∩(−K+
), and rank(JXJ) ≤ r.
7. Dealing with Spherical Constraints
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We now want to place n points on a sphere:
kxi k = R.
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We assume the center to be the (n + 1)th point xn+1 so that
kxi − xn+1 k2 = R2 , ∀ i = 1, . . . , 2.
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The formulation of optimization problems with spherical
constraints takes the following form:
min / max
s.t.
f (Y )
n+1
Y ∈ Shn ∩ (−K+
)
rank(JXJ) ≤ r
Y1(n+1) = Yi(n+1) , i = 2, . . . , n.
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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
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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 − pj k = 1 ∀ (i, j) ∈ E.
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Def. If furthermore,
kpi k = kpj k ∀ 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
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Finding the smallest radius of a hypersphere representation
(Lovász (’09), Silva and Tuncel (’10))
th (G) := min t
s.t. diag(X) = te
Xii − 2Xij + Xjj = 1, ∀ (i, j) ∈ E
n , t ∈ <.
X ∈ S+
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It is known
2th (G) +
I
1
= 1.
ϑ(G)
EDM formulation:
min Y1(n+1)
s.t.
n+1
Y ∈ Shn ∩ (−K+
)
Yij = 1 ∀ (i, j) ∈ E
Y1(n+1) = Yi(n+1) , i = 2, . . . , n.
9. Lovász-theta Function
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Def. An orthonormal representation of G is a system
{p1 , . . . , pn } of unit vectors in a Euclidean distance space
such that
hpi , pj i = 0
∀ (i, j) 6∈ E.
Theorem 5, Lovász (’79): Let (p1 , . . . , pn ) range
over all orthonormal representations of G and d over
all unit vectors. Then
ϑ(G) = max
n
X
(hd, pi i)2 .
i=1
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SDP formulation:
ϑ(G) = max hJ, Xi
s.t. hI, Xi = 1
Xij = 0, ∀ (i, j) ∈ E
X 0.
From Projection to Euclidean Distance
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We have
kd − pi k2 = kdk2 + kpi k2 − 2hd, pi i = 2 − 2hd, pi i.
Hence
2
1
1 − kd − pi k2 .
4
Under the condition (part of Schrijver’s ϑ0 function):
(hd, pi i)2 =
I
hd, pi i ≥ 0,
we have
2
1
kd − pi k2
4
This leads to the following EDM problem
2
P
p(G) := min 14 ni=1 kd − pi k2
max (hd, pi i)2
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⇐⇒
min
s.t. kpi k = 1; kdk = 1;
hpi , pj i = 0 ∀ (i, j) ∈ E.
A Quantity that may be interesting
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For a given graph, define the quantity q(G) such that
p
p
√
p(G) + q(G) = n.
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Let SOL(G) denote the solution set of the SDP of ϑ function.
Define
b(ϑ)
τϑ := √ ,
n
where
b(ϑ) := max
Pn √
i=1
Bii
s.t. B ∈ SOL(G).
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For vertex-transitive graphs
τϑ = 1.
Bound that measures Distortion
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Define
rϑ :=
p
n/ϑ(G)
and
1p
(r − 1)2 + 2r(1 − τ ).
τ
Define the distortion constant dϑ by
tϑ := r −
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dϑ := tϑ τϑ
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Claim (Bound of Distortion):
d2ϑ ϑ(G) ≤ q(G) ≤ ϑ(G).
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Remark: dϑ hard to calculate. But for vertex-transitive
graphs, we have
dϑ = 1.
Is Triangle Inequality `2 Metric?
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One can add triangle inequalities to SDP to strengthen ϑ
function:
Xik + Xjk ≤ Xij + Xkk
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(i, j, k ∈ V ).
Let X 0 admit the Gram representation:
X = P T P.
Therefore,
kpi − pk k2 = Xii + Xkk − 2Xik .
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`2 -metric:
kpi − pk k2 + kpj − pk k2 ≥ kpi − pk k2
which implies
1
Xik + Xjk ≤ Xik + (Xkk + Xjj ).
2
A Wild Guess
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The close τϑ to 1, the less room that adding cut (triangle)
inequalities can strengthen ϑ(G).
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Example is vertex-transitive graphs.
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We can measure this by computing the ratio:
ϑ(G)
q(G)
Both are convex problems.