Matrix Completion,
G = ([n], E)
Free Resolutions, and
S →
R ⊕R
π ∶�
(a ) � (a , . . . , a ) ⊕ (a ∶ {i, j} ∈ E)
Sums of Squares
Matrix Completio
Throughout, we fix a simple graph
This encodes the coordinate projection
n
G
ij
n
��
nn
.
�E
ij
n : real vector space of symmetric n × n matrices.
S
Rainer Sinn (Georgia Institute of Technology)
NB: Think of πG (a i j) as a partial matrix.
�
�
�
�
Joint
withLet G be the four cycle.
Example.
Grigoriy Blekherman (Georgia� aInstitute
of
Technology)
a
a
a
�� �� �� �� � � a�� a�� ∗
� a�� a�� de
� a�� a�� a��
Mauricio Velasco (Universidad
Andes)
a�� los
a�� �
�
�
�
a�� �
∗ �
�
πG �
=�
�
�
a
a
a
a
∗
a
a
a
� �� �� �� �� � �
�� �� �� �
�a�� a�� a�� a��� �a�� ∗ a�� a���
Throughout, we fix a simple graph G
= ([n],Completion
E).
Matrix
This encodes the coordinate projection
Throughout,Swe
G
([n],Completion
E).
n fix
Matrix
→ a simple graph
Rn ⊕
R=�E
πG ∶ � the coordinate projection
This encodes
(a i j) � (a�� , . . . , a nn ) ⊕Matrix
(a i j∶ {i,Completion
j} ∈ E)
Throughout,Swe
a simple graph
G R=�E
([n], E).
n fix
n⊕
→
R
n
S
: real
space
symmetric
nG ×=n([n],
matrices.
Throughout,
fix
a of
simple
graph
E).
πG vector
∶ � we
This
encodes
the
coordinate
projection
(a
. . , projection
a nnmatrix.
) ⊕ (a i j∶ {i, j} ∈ E)
��a, .partial
jG)(a�
NB:
of(aπithe
)
as
ThisThink
encodes
coordinate
ij
Throughout, we fix a simple graph
n
n
�E
S space
→ of symmetric
R ⊕
R n matrices.
�
S�n : real
vector
n
×
n
n
�E
This encodes the coordinate projec
πG ∶ � S →
R
⊕
R
. . , a nnmatrix.
) ⊕ (a i j∶ {i, j} ∈ E)
��a, .partial
πG ∶ �of(aπiGj)(a�i j)(a
NB: Think
as
(a i j) � (a�� , . . . , a nn ) ⊕ (a i j∶ {i, j} ∈ E)
n →
n
S
R
�
πG ∶ �
S�nn : real vector
space of symmetric n × n matrices.
(a i j) � (a�� , . . . , a nn ) ⊕
�
�
S
:
real
vector
space
of
symmetric
n
×
n
matrices.
NB: Think of πG (a i j) as a partial matrix.
Matrix Completion
NB: Think�of πG (a i j) as a partial matrix.
Sn : real vector space of symmetric
�
Example.
� Let G be the four cycle.
�
NB: Think of πG (a i j) as a partial m
Throughout, we fix a simple graph G = ([n], E).
�
�
a�� coordinate
a�� a�� � projection
� a�� the
� a�� a�� ∗ a�� �
This encodes
Example.
G beatheafour�cycle.
��aLet
� an�� a���Ea�� ∗ �
�
a
��
��
��
��
n
�
�
�R ⊕ R
�
=
� πG �� S →
� �∗ a a a �
πG ∶��aa��
a
a
a
��
��
��
��
a
a
a
a)��⊕a(a
aj}��
�� i j )��� ��
��. .�. , a�
�� i j ∶∗��
�� ∈� E)
(a
(a
,
{i,
nn
��
�
Example.
G��beaathe
four
cycle.
�
�aaLet
�
�
�
a
a
a
∗
a
a
��
��
��
��
��
��
a
a
a
a
a
∗
�� G��be the
�� four
�� �cycle.
�� �� ��
Example.
Let
�
�
�
n
π
=
S : realG vector
of
symmetric
n
×
n
matrices.
� a aspace
�
�
�
a
a
∗
a
a
a
��
��
��
��
��
��
��
a
∗
�
�
�
�
�� ��a G��-partial
�� matrix
�� to��a positive
�� semidefinite symmetric
Goal:
Complete
matrix
(and
control
Example.
Let
G
be
the
four cycle.
NB: Think
of
π
(a
)
as
a
partial
matrix.
a
a
a
a
a
a
∗
a
�
�
�
�
G
i
j
�� a��
�� a��
�� a��
�� � � a��
�� a∗�� a�� a∗
��
�a��
�
��
�� �� �� � � �� �� �� ���
the
rank
of
the
completion).
�
�
�
=�
�a�� a�� a�� a�� �
� πG �
a
a
a
∗
�
�
�
��
��
��
�
�
�
�
� a�� a�� a�� a�� � � a��
a
a
a
a
∗
a
a
a
π
=
�
�
�
�
��
��
��
��
��
��
��
G
�
�
�
�
Goal: Complete
a aG��-partial
matrix
to��a apositive
semidefinite symmetric
matrix (and control
a
a
a
∗
a
a
�
�
�
�
��
��
��
��
��
�
� a��
Geometry
of
positive
semidefinite
matrix
completion
a
�a�� a�� a�� a��� �a�� ∗ a�� a���
�� a�� a�� a�� �
�
�
�
π
=
the rank�ofa��
theacompletion).
�
�
�
a
a
a
∗
a
a
G
�a a a a � � ∗
�� �� ��
��
�� ��
� �� �� �� �� � �
�
�
n
Goal: Describe
Completethe
a Gimage
-partialofmatrix
to aSpositive
semidefinite
symmetric
matrix
(and
control
the
cone
of positive
semidefinite
quadratic
forms
under
the
�
a
a
a
a
≥�
��
��
��
��� � a��
Geometry
of
positive
semidefinite
matrix
completion
Goal: Complete a G -partial matrix to a positive semidefinite symmetric matrix (and control
the rank of πthe
projection
. completion).
G
Example.
be the four cycle.
the
rank of Let
theGcompletion).
n
a �G,under
S
cone
of image
quadratic
forms
x i positive
x j such that
p j ≥Complete
� for all
(p
.-partial
. . , p nthe
) matrix
∈
∑i,Sjna i jof
∑i, j a iGoal:
j p iquadratic
Goal:
Describe
the
of the
cone
semidefinite
forms
≥�: convex
Goal: Complete a G -partial matrix to a positive semidefinite symmetric matrix (and control
Goal: Complete a G -partial matrix to a positive semidefinite symmetric matrix (and control
the rank of the completion).
the rank of the completion).
Geometry of positive semidefinite matrix completion
Geometry of positive semidefinite matrix completion
n of positive semidefinite quadratic forms under the
Goal: Describe the image of the cone S≥�
n of positive semidefinite quadratic forms under the
Goal: Describe the image of the cone S≥�
projection πG .
projection
πG .
n
S≥�
cone of quadratic forms ∑ i, j a i j x i x j such that ∑ i, j a i j p i p j ≥ � for all (p� , . . . , p n ) ∈
n :: convex
S≥�
convex
cone of quadratic forms ∑ i, j a i j x i x j such that ∑ i, j a i j p i p j ≥ � for all (p� , . . . , p n ) ∈
n
Rn .
R .
Theorem (Diagonalization of Quadratic Forms). A quadratic form q ∈ R[x� , . . . , x n ] is positive
Theorem (Diagonalization of Quadratic Forms). A quadratic form q ∈ R[x� , . . . , x n ] is positive
semidefinite if and only if it is a sum of squares of linear forms after a change of basis.
semidefinite if and only if it is a sum of squares of linear forms after a change of basis.
n ) a cone of sums of squares?
Question: Is πG (S≥�
n ) a cone of sums of squares?
Question: Is πG (S≥�
Stanley-Reisner Ideals
Stanley-Reisner Ideals
Recall G = ([n], E) is fixed.
Recall G = ([n], E) is fixed.
Let IG = �x i x j∶ {i, j} ∈� E� ⊂ R[x� , . . . , x n ]
Let IG = �x i x j∶ {i, j} ∈� E� ⊂ R[x� , . . . , x n ]
Note: The degree � part of IG is the kernel of πG .
Note: The degree � part of IG is the kernel of πG .
Then IG is the Stanley-Reisner ideal of the clique complex of the graph:
Then IG is the Stanley-Reisner ideal of the clique complex of the graph:
simplicial complex ∆ on [n], where H ∈ ∆ if and only if the induced subgraph of G on the
simplicial complex ∆ on [n], where H ∈ ∆ if and only if the induced subgraph of G on the
vertices in H is complete.
vertices in H is complete.
So a monomial x H = ∏ i∈H x i is in IG if and only if G�H is not a complete graph, i.e. there are
So a monomial x H = ∏ i∈H x i is in IG if and only if G�H is not a complete graph, i.e. there are
i, j ∈ H such that {i, j} ∈� E . But then x i x j ∈ IG and x i x j�x H .
i, j ∈ H such that {i, j} ∈� E . But then x i x j ∈ IG and x i x j�x H .
Subspace Arrangement
Subspace Arrangement
Question: Is πG (S≥�
) a cone of sums of squares?
n
Question:
(S
aa cone
of
sums
of
squares?
n )
≥�
Question: Is
Is π
πG
(S
)
cone
of
sums
of
squares?
G ≥�
Stanley-Reisner Ideals
Stanley-Reisner Ideals
Stanley-Reisner Ideals
Recall G = ([n], E) is fixed.
Recall G = ([n], E) is fixed.
Let IG = �x i x j∶ {i, j} ∈� E� ⊂ R[x� , . . . , x n ]
Let
I G
= �x
∈�fixed.
E� ⊂ R[x� , . . . , x n ]
i x j ∶ {i,
Recall
=degree
([n],
E)�j}is
Note:GThe
part
of IG is the kernel of πG .
Note:
The
degree
�j}part
of⊂IR[x
is �the
kernel
of πG .
G
Let
I
=
�x
x
∶
{i,
∈
�
E�
,
.
.
.
,
x
]
n
i
G
j
Then IG is the Stanley-Reisner ideal of the clique complex of the graph:
Then
IIG
is the
ideal
clique complex
of
graph:
Then
the Stanley-Reisner
Stanley-Reisner
ideal of
ofHthe
the
complex
of the
the
graph: subgraph of G on the
G is complex
simplicial
∆ on [n], where
∈ clique
∆ if and
only if the
induced
simplicial
complex
∆
on
[n]
,, where
H
∈∈ ∆
if
and
only
if
the
induced
subgraph
of
G
on
the
simplicial
complex
∆
on
[n]
where
H
∆
if
and
only
if
the
induced
subgraph
of
G
on
the
vertices in H is complete.
vertices
in
vertices
in H
H is
is complete.
complete.
So a monomial
x H = ∏i∈H x i is in IG if and only if G�H is not a complete graph, i.e. there are
So
aa monomial
xx H =
xx i is
in
IIG if
and
only
if
G�
is
not
aa complete
graph,
i.e.
there
are
∏
H
i∈H
So
monomial
=
is
in
if
and
only
if
G�
is
not
complete
graph,
i.e.
there
are
∏
i thenGx i x j ∈ IG and x i xH
i, j ∈ H such that H{i, j} ∈�i∈H
E . But
j �x H .
i,
and x x �x ..
i, jj ∈∈ H
H such
such that
that {i,
{i, j}
j} ∈∈�� E
E .. But
But then
then xx ii xx jj ∈∈ IIG
G and x ii x jj �x H
H
Subspace
Arrangement
Note: The degree � part of IG is the
kernel
of
π
G.
Subspace Arrangement
The quadratic form �x i x j corresponds to the symmetric matrix E i j + E ji .
Set XG = V(IG ) ⊂ Pn−�
, the subscheme associated with the Stanley-Reisner ideal IG .
n−�
Set XG = V(IG ) ⊂ P , the subscheme associated with the Stanley-Reisner ideal IG .
Subspace Arrangement
XG = � span{x i ∶ i ∈ K},
XG = K⊂G
� span{x
i ∶ i ∈ K},
n−�
�
Set XG = V(I
, the subscheme associated
with the Stanley-Reisner ideal IG .
K⊂G
G) ⊂ P
�
Subspace Arrangements
Subspace Arrangements
Throughout, we fix a simple graph
Set XG = V(IG ) ⊂ Pn−�, the subscheme associated with the Stanley-Reisner
ideal
IGcoordinate
.
This encodes
the
proje
Set XG = V(IG ) ⊂ Pn−�, the subscheme
associated
with the Stanley-Reisner ideal
IG .
Subspace
Arrangements
n
n
S
→
R
Subspace Arrangements
πG ∶ �
XG = � span{e
∶
i
∈
K},
(a i j) � (a�� , . . . , a nn )
i
n−�
Set XX
⊂ P ,∶ the
subscheme associated with the Stanley-Reisner ideal IG .
G ==V(I
G )span{e
K⊂G
i
∈
K},
�
n−�
Set XGG= V(IG ) ⊂ P i , the subscheme associated with the Stanley-Reisner
idealspace
IG . of symmetric
n : real vector
S
K⊂Gover all complete subgraphs of G .
where K runs
NB: Think of πG (a i j) as a partial m
whereXG
K =runs
over
all
complete
subgraphs
of
G
.
� span{e i ∶ i ∈ K},
Example.
Let
be thei four
cycle.
�
�
XG = K⊂G
∶ i ∈ K},
� Gspan{e
Example.
Let
fourXcycle.
K⊂G
Then IG = �x
xG
, xbe� xthe
� and
is the union of four lines in P�.
�
�
�
G
where K runs over all complete subgraphs of G .
�.
Then
I
=
�x
x
,
x
x
�
and
X
is
the
union
of
four
lines
in
P
�
�
�
�
G
G
where K runs over all complete subgraphs of G .
Observation: The projection πG is the degree � part of the map of coordinate
rings R[x� , . . . , x n ] →
�
�
Example. Let G be the four cycle.
Observation:
projection
πG is the degree � part of the map
of coordinate rings R[x� , . . . , x n ] →
R[X
]
. Let The
Example.
G
be
the
four
cycle.
�
G
Then IG = �x� x� , x� x�� and XG is the union of four lines in P .
�.
R[X
ThenGI]G. = �x� x� , x� x�� and XG is
the
union
of
four
lines
in
P
Example.
Let
G beisthe
four
cycle.
n ) of the cone of positive semidefinite
Proposition.
The
image
π
(S
quadratic
forms
the
cone
G π≥�is the degree � part of the map of coordinate rings R[x , . . . , x ] →
Observation: The projection
n
G ) of the cone of positive semidefinite quadratic forms is�the cone
n
Proposition.
The
image
π
(S
Σ XG of].all quadratic
forms inG R[X
are sums
of of
linear
forms.� arings
Observation:
The projection
π≥�
is
degree
� partofofsquares
the map
coordinate
R[x
a
a
� that
�
� a�
n] →
G ]the
� , . .a.��, x
G
��
��
��
R[X
G
Σ XG Gof].all quadratic forms in R[XG ]� that are sums of squares of linear forms.� a a a a � � a
R[X
�� �� �� �� � � �
�
π
=�
n
G � a forms
�
Proposition. TheSums
imageof
πGSquares
(S≥�
)
of
the
cone
of
positive
semidefinite
quadratic
is
the
cone
a�� a�� a�� � � ∗
and
Nonnegative
Polynomials
� ��forms
n ) of the
Proposition.
The
image
π
(S
cone
of
positive
semidefinite
quadratic
is the cone
G
≥�
Σ XG of all quadratic
forms
in
R[X
]
that
are
sums
of
squares
of
linear
forms.
Sums of Squares
G � and Nonnegative Polynomials
�a�� a�� a�� a��� �a�
Σ XG of all quadratic forms in R[XG ]� that are sums of squares of linear forms.
Observation:
Every sum of squares of linear forms is a nonnegative quadric.
Goal:
Complete a G -partial matri
Observation:r Every
sumof
of linear
forms is a nonnegative
quadric.
Sums
Squares
and Nonnegative
Polynomials
rof squares
�
the rank of the completion).
of
and
Nonnegative
Polynomials
r ℓ�Sums
r (ℓSquares
q(p) = �
(p)
=
(p))
�
i
�i (p) =
�
q(p)
=
ℓ
(ℓ
(p))
�
�
i=�
i=�
i
Observation: Every
sum of squares
of linear forms is a nonnegative quadric.
i
i=�Every sumi=�
Observation:
of squares of linear forms is a nonnegative quadric. Geometry of po
r of nonnegative quadrics.
We write PXG rfor the cone
rfor
r (ℓ
We write
of inonnegative
quadrics.
q(p)P=XG�
ℓ�ithe
(p)cone
=�
(p))�
Σ XG of all quadratic
forms in R[X
]of� that
are sums of squares
of linear
forms.
n G
Matrix
Completion
Proposition.
The
image
π
(S
)
the
cone
of
positive
semidefinite
quadratic
forms is the cone
R[X
]
.
G ≥�
G
Σ XG of all quadratic forms in R[X
G ]� that are sums of squares of linear forms.
n
Throughout,
we
fix acone
simple
graph
G
= ([n], E).quadratic forms is the cone
Proposition. The image
πG (S≥�)Summary
of the
of
semidefinite
ofpositive
the setup
This encodes
coordinate
Σ XG of all quadratic forms
in R[XG the
]� that
are sumsprojection
of squares of linear forms.
Summary
of�xthe
setup
Let G = ([n], E) be a simple graph
and
let
I
=
x
∶
{i,
∈� E� be the Stanley-Reisner ideal of
i R
G
j n ⊕ j}
Sn →
R�E
the clique complex of G . πG ∶ � (aSummary
the
setup
) n�let
(aI��nG, =
.of
. �x
. ,�E
ai xnn
⊕ j}
(a∈�i j∶E�
{i,bej}the
∈ E)
Let G = ([n], E) be a simple graphi jand
Stanley-Reisner ideal of
j ∶ ){i,
Then the coordinate projection πG ∶ S → R ⊕ R is the same as the degree � part of the map
the clique complex of nG .
Matrix
Completion
S
:
real
vector
space
of
symmetric
n
×
n
matrices.
R[x
, .=. .([n],
, x n ] E)
→ be
R[X
] = R[x
, x n ]�I
coordinate
rings.
Let G
aG
simple
graph
letGIGofn=homogeneous
�xRi x�Ej∶ {i,
j} ∈� E�
be the Stanley-Reisner
ideal of
� , .π. . and
Then� the coordinate
projection
∶
S
→
R
⊕
is
equal
to
the
degree
�
part
of
the
map
G (a ) as a partial matrix.
n
NB:
Think
of
π
Therefore,
the imageofπGGThroughout,
ofgraph
squares
Σ X([n],
⊂ R[X
]� on XG .
G cone
i j fixofa sums
the clique complex
.(S≥�) is the
Grings.
G
we
simple
G
=
E)
.
R[x� , . . . , x n ] → R[X
]
=
R[x
,
.
.
.
,
x
]�I
of
homogeneous
coordinate
n Gn
�
G
n
�E is the same as the degree � part of the map
�
�
Then the coordinate projection
π
∶
S
→
R
⊕
R
n
G
the coordinate
Therefore, the image πGThis
(S≥�encodes
) is the cone
of sums of projection
squares Σ XG ⊂ R[XG ]� on XG .
R[x� , . . . , x n ] → R[X
homogeneous coordinate
rings.
Sums
ofR[x
Squares
and
Nonnegative
Polynomials
� , . . . , xnn ]�I
G] =
G of
n ⊕ R�E
n
S
→
R
Therefore, the image πG (S≥�π) is∶ �
the cone of sums of squares Subspace
Σ XG ⊂ R[XGArrangements
]� on XG .
G
Sums
Squares
and
Nonnegative
� (aofi j)linear
� (a
) ⊕ (aPolynomials
j} ∈ E)
� of
�� , . . .is, aann
i j ∶ {i, quadric.
Observation: Every
sum
of squares
forms
nonnegative
n−�, the subscheme
rn : real
Sums
of
Squares
and
Nonnegative
S
vector
space
of
symmetric
n ×Polynomials
n matrices.
Set
X
=
V(I
)
⊂
P
associated with the Stanley-Re
G
G
Observation: Every
sum
of
squares
of
linear
forms
is a nonnegative
quadric.
�
�
r
Let G be the four cycle.
q(p) = � ℓ i (p)Example.
= NB:
�(ℓThink
i (p))of πG (a i j ) as a partial matrix.
r
r
i=�
i=�
Observation: Every
of linear
forms
isaa��nonnegative
quadric.
�a��
a
a
a
a
∗
a
�
�
�
�
�
�of(ℓsquares
��
��
��
��
��
q(p) = � ℓ�i (p)sum
=�
(p))
i
XG = �quadrics.
span{e
∶ i ∈ K},
i
We write PXGi=�
of
nonnegative
�
�
�
rfor the cone
r
a�� � � a�� a�� a�� ∗ �
i=� � a�� a�� a��
�
K⊂G
�
=�
G i�(p))
�
�
q(p) = � ℓ�i (p) = �π(ℓ
a
a
a
a
∗
a
a
a
�
�
�
� . of
��
��
��
��
��
��
��
We
write PXGi=�
for the cone
of
nonnegative
quadrics.
Proposition
(Obvious
necessary
condition).
Assume
π
(A)
∈
Σ
Then
every completely
where
K
runs
over
all
complete
subgraphs
G
.
G
X
i=�
G
� �a��� a�� a�� a��� �a�� ∗ a�� a���
specified
of A is positivequadrics.
semidefinite.
We
writesymmetric
PXG (Obvious
for thesubmatrix
cone
of nonnegative
Proposition
necessary
condition).
G (A) ∈ Σ XG . Then every completely
Example. Let G beAssume
the fourπcycle.
Goal:
Complete
aGGbe
-partial
matrix
to a positive semidefinite symmetric
ma
Example.
Let
the
four
cycle.
�
specified
symmetric
submatrix
of
A
is
positive
semidefinite.
Proof.
Σ
⊂
P
.
�
Then IG = �x� x� , x�Assume
x � and π
XGG(A)
is the∈union
four every
lines incompletely
P.
XG
XG
Proposition
(Obvious
Σ XG .ofThen
thenecessary
rank of thecondition).
completion). �
ais��positive
a�� a��semidefinite.
a�� � � a�� a�� ∗ a�� �
specified
submatrix
of �
AThen
Proof.
Σ Xsymmetric
⊂
P
.
Example.
Let
G
be
the
four
cycle.
X
Observation:
The
projection πG is the degree � part of the map of�coord
G
G
� aGeometry
�positive
� a�� a��semidefinite
a�� a�� aof
a�� ∗ �
matrix completio
��
��
�
�
�
�
R[X
]
.
π
=
G
G
Proof.
Σ XG ⊂
.a��the
�
� �∗ a a a �
a��four∗cycle.
a���
Example.
LetPX
G�
Then
�
Gbe
a
a
a
a
� �� �� �� �� � �
�� �� �� �
� a�� Goal:
n∗n) a
a�� a��
∗��
�
�
� of
Describe
the
image
of
the
cone
S
of
positive
semidefinite
quadraticquf
a
a
a
a
a
a��
Proposition.
The
image
π
(S
of
cone
positive semidefinite
�
�
��
��
��
��
��
��the
G
≥�
Example.
G
be
the
four
cycle.
Then
πG (a iLet
)
=
.
≥�
a
a
∗
a
�
�
j
∗�� a �� a a ��
G �
XG
Sums
Sums of
of Squares
Squares and
and Nonnegative
Nonnegative Polynomials
Polynomials
Sums of Squares and Nonnegative Polynomials
Observation:
Observation: Every
Every sum
sum of
of squares
squares of
of linear
linear forms
forms is
is aa nonnegative
nonnegative quadric.
quadric.
Observation:r Every sumrof squares of linear forms is a nonnegative quadric.
r �
r
��
�
r
r
q(p)
=
ℓ
(p)
=
(ℓ
(p))
�
�
i
q(p) = � ℓ�ii (p) = �(ℓ i (p))�
q(p) = �
i=�
i=�
i=� ℓ i (p) = �
i=� (ℓ i (p))
i=�
i=�
i=�
i=�
Matrix Co
We
for
the
cone
of
nonnegative
quadrics.
We write
write P
PX
for
the
cone
of
nonnegative
quadrics. Throughout, we fix a simple graph G = ([n], E)
G
X
X
G
G
We write PXG for the cone of nonnegative quadrics.
This encodes
the
coordinate
projection
Proposition
(Obvious
necessary
condition).
Assume
π
(A)
∈
Σ
.
Then
every
completely
X
Proposition (Obvious necessary condition). Assume πG
(A)
∈
Σ
.
Then
every
completely
G
(A)
∈
G
X
G
G
X
Proposition
(Obvious
necessary
condition).
Assume πG (A) ∈n Σ XGG . Then every completely
n ⊕ R�E
specified
symmetric
submatrix
of
A
is
positive
semidefinite.
S
→
R
specified
symmetric
submatrix
of
A
is
positive
semidefinite.
specified
specified symmetric
symmetric submatrix
submatrix of
of A
A is
is positive
positive semidefinite.
semidefinite. πG ∶ �
(a i j) � (a�� , . . . , a nn ) ⊕ (a i j∶ {i, j}
Proof.
Σ
⊂
P
..
�
X
X
Proof.
Σ
⊂
P
�
G
G
X
X
Proof.
Σ
⊂
P
.
�
G
G
X
Proof. Σ X
⊂
P
.
�
n
G
XG
X
G
G
S : real vector space of symmetric n × n matrice
Example.
Let
G
be
the
four
cycle.
Then
Example.
Let
G
be
the
four
cycle.
NB: Think of πG (a i j) as a partial matrix.
Example. Let G be the four cycle. Then
Then
�
�
aa��
aa��
∗
aa��
�
�
∗
�
�
��
��
��
a�� a�� ∗ a�� �
�
�
�
aa��
a
a
∗
�
�
��
��
a
a
∗
�
�
��
��
��
a
a
a
∗
π
(a
)
=
..
�
�
��
��
��
G
i
j
�
�
π
(a
)
=
G
i
j
�
�
∗
aa��
aa��
aa��
πG (a i j) = �
.
�
∗
�
�
��
��
��
∗
a
a
a
�
�� �� �� �
�
�
�
�
a
∗
a
a
��
��
��
�
�
∗ a�� aa��
�aa��
�� ∗ a��
���
The
four
fully
specified
submatrices
correspond
to
the
four
maximal
cliques
of
G
,, namely
the
edges.
Example.
Let
G
be
the
four
cycle.
The
four
fully
specified
submatrices
correspond
to
the
four
maximal
cliques
of
G
namely
the
The four fully specified submatrices correspond to the four maximal cliques of G , namely the edges.
edges.
Question:
When
is
this
obvious
necessary
condition
sufficient?
� a�� a�� a�� a�� � � a�� a�� ∗ a��
Question:
When
is
this
obvious
necessary
condition
Question: When is this obvious necessary condition sufficient?
sufficient?
� a�� a�� a�� a�� � � a�� a�� a�� ∗
� �
πG �
�a a a a � = � ∗ a a a
� �� �� �� �� � �
�� �� ��
Two
long
lost
friends
Two
long
lost
friends
Two long lost friends�a�� a�� a�� a��� �a�� ∗ a�� a��
Recall:
G
=
([n],
E)
is
aa simple
graph.
Recall:
G
=
([n],
E)
is
simple
Goal: Complete a G -partial matrix to a positiv
Recall: G = ([n], E) is a simple graph.
graph.
Proof. Σ XG ⊂ PXG .
Question:
is this
necessary condition sufficient?
Recall: G = When
([n], E)
is a obvious
simple graph.
Example.
Letchordal
G be theif four
cycle.
Then
Two
lost
A
graph G is
every
cycle
in G long
of length
at friends
least � has a chord.
� spac
Sn : real vector
NB: Think of πG (a i
�
�
Two long lost
friends
a
a
∗
a
�
�
��
��
��
Theorem
(Agler-Helton-McCullough-Rodman)
and
Theorem
and (Paulsen-Power-Smith).
(Paulsen-Power-Smith). The
The obvious
obvious necnecRecall: G =(Agler-Helton-McCullough-Rodman)
([n], E) is a simple graph.
�
�
a��positive
a�� a��
∗ �
essary
condition
for
semidefinite
matrix
completion
is
sufficient
(equivalently,
Σ
=P
))
�
X
X
essary
condition
for
positive
semidefinite
matrix
completion
is
sufficient
(equivalently,
Σ
=
P
A
graph
G=iisj([n],
cycle
in
G
of
length
at
least
�
has
a
chord.
πGG(a
)chordal
= �E) isifaevery
.
G
G
X
X
Recall:
simple
graph.
� G � G
∗ a�� a�� a�� �
�
�
if
and
only
if
G
is
chordal.
if
only
is chordal.
A and
graph
G ifisGchordal
if∗every
cycle
in G of length at least � has a chord.
�
�
a
a
a
Theorem (Agler-Helton-McCullough-Rodman)
and (Paulsen-Power-Smith). The obvious nec��
�� ��
Theorem
(Blekherman-Sinn-Velasco).
Every
nonnegative
is
sum
on
X
(in
Example.
Let
G be
G
Theorem
(Blekherman-Sinn-Velasco).
Every completion
nonnegative
quadric
is a
a(equivalently,
sum of
of squares
squares
on
X
(in
essary
condition
for positive semidefinite matrix
isquadric
sufficient
Σobvious
=
P
)
G
Theorem
(Agler-Helton-McCullough-Rodman)
and
(Paulsen-Power-Smith).
The
necX
X
G edges.
G
The
four Σ
fully specified
submatrices
correspond
to the four maximal cliques of G , namely the
symbols
=
P
)
if
and
only
if
I
is
�
-regular.
X
symbols
ΣX
= Pchordal.
) if and only if IG
if
and only
XifG
XG
G is �-regular.
essary
condition
matrix completion is sufficient (equivalently, Σ XG =�PaX��G ) a�� a
GG is for
G positive semidefinite
Question:
is thisThe
obvious
necessary
condition
sufficient?
if
and only(Fröberg).
ifWhen
G is chordal.
� a��
Theorem
monomial
ideal
I
is
�
-regular
if and
only ifisGa issum
chordal.
a�� a
G
�Theorem
-regular:(Blekherman-Sinn-Velasco).
The minimal free resolution is Every
linear:nonnegative
quadric
of squares on
X
(in
�
G
πG �
a�� a
� a��
symbols
=
P
)
if
and
only
if
I
is
�
-regular.
φΣt+�
φ
φ
φ
Theorem
(Blekherman-Sinn-Velasco).
Every
nonnegative
quadric
is
a
sum
of
squares
on
X
(in
t
t−�
�
X
X
G
G
Two
long
� ��→G Ft �→G Ft−� ��→ � �→
F� →
I → �lost friends
�
a�� a�� a
symbols Σ XG = PXG ) if and only if IG is �-regular.
Theorem (Fröberg). The monomial ideal IG is �-regular if and only if G is chordal.
Theorem
monomial
I is �-regular if and only if G is chordal. Goal: Complete a
Recall:
G =(Fröberg).
([n], E) is The
a simple
graph.ideal
Theorem
(Fröberg).
The
monomial
ideal IG
G is �-regular if and only if G is chordal.
the rank of the com
A graph G is chordal if every cycle in G of length at least � has a chord.
Theorem (Eisenbud-Green-Hulek-Popescu). The ideal I is �-regular if and only if V(I) is a linearly
joined
arrangement of subspaces if and only and
if V(I)
is small.
Theorem
(Agler-Helton-McCullough-Rodman)
(Paulsen-Power-Smith).
The obvious necGeom
essary condition for positive semidefinite matrix completion is sufficient (equivalently, Σ XG = PXG )
References
if and only if G is chordal.
Goal: Describe the
projection πG .
Blekherman,
Sinn,
Velasco.
Small
Schemes
and
Sums
of
Squares,
to
appear
on
arXiv
soon.
Theorem (Blekherman-Sinn-Velasco). Every nonnegative quadric is a sum of squares
on
XG (in
n
S≥�: convex
cone o
Agler,
Positive semidefinite matrices with a given sparsity
patsymbolsHelton,
Σ XG = PMcCullough,
if IG is �-regular.
n
XG ) if and onlyRodman.
R .
tern, Proceedings of the Victoria Conference on Combinatorial Matrix Analysis (Victoria, BC, ����)
Theorem
(Fröberg). The monomial ideal IG is �-regular if and only if G is chordal.
(����)
Theorem (Diagon
Fröberg.On Stanley-Reisner rings, Topics in algebra, Part � (Warsaw, ����) (����) semidefinite if and o
Paulsen, Power, Smith. Schur products and matrix completions, Journal of Functional Analysis
Question: Is π (S
Theorem
Theorem (Eisenbud-Green-Hulek-Popescu).
(Eisenbud-Green-Hulek-Popescu). The
The ideal
ideal II is
is ��-regular
-regular if
if and
and only
only if
if V(I)
V(I) is
is a
a linlinTheorem
(Eisenbud-Green-Hulek-Popescu).
The
ideal
I
is
�
-regular
if
and
only
if
V(I)
is
a
linTheorem
(Eisenbud-Green-Hulek-Popescu).
The
ideal
I
is
�
-regular
if
and
only
if
V(I)
is
a
linearly
joined
arrangement
of
subspaces
if
and
only
if
V(I)
is
small.
early joined arrangement of subspaces if and only if V(I) is small.
early
early joined
joined arrangement
arrangement of
of subspaces
subspaces if
if and
and only
only if
if V(I)
V(I) is
is small.
small.
Minimal Free Resolutions and Matrix Completion
Minimal
Minimal Free
Free Resolutions
Resolutions and
and Matrix
Matrix Completion
Completion
The
is
The dual
dual convex
convex cone
cone to
to Σ
ΣX
is
G
X
G
The
dual
convex
cone
to
Σ
is
X
The dual
convex
cone
to
Σ
G
XG is
∨
∗
∨
∗
Σ
=
{ℓ
∈∈ R[X
]]�∗∶∶ ℓ(
ff )) ≥
�� for
}.
G
X
Σ∨X
=
{ℓ
R[X
ℓ(
≥
for all
all ff ∈∈ Σ
Σ
}.
G
G
X
G
�
X
G
G
Σ
=
{ℓ
∈∈ R[X
]]∗� ∶∶ ℓ(
ff )) ≥
�� for
all
ff ∈∈ Σ
}.
G
X
G }.
Σ∨X
=
{ℓ
R[X
ℓ(
≥
for
all
Σ
G
G
X
�
XG
∨G � Σ∨∨ . So there is an extreme ray ℓ ∈ Σ
∨
Convex
Biduality:
If
Σ
�
P
,
then
dually
P
Convex Biduality: If Σ X
Σ∨XXG. So there is an extreme ray ℓ ∈ Σ XXGG,
XG
XGG, then dually PX
G � PX
XGG �
∨
G . So there is an extreme ray ℓ ∈ Σ X ,
Convex
Biduality:
If
Σ
�
P
,
then
dually
P
�
∨ � Σ
∨
X
X
G
G
X
X
Convex
Biduality:
If
Σ
�
P
,
then
dually
P
Σ
,
which
.. XG
G
G . So there is an extreme ray ℓ ∈ Σ XG
XG
which is
is not
not in
in PX
G
X
X
G
X
G
G
G.
which
is
not
in
P
∗ with a space of symmetric matrices:
X
⊥ R[xI ⊥
∗]∗= (R[x
G. G
∗Gwith
which
not in PR[X
We
canisidentify
identify
=
(R[x
,
.
.
.
,
x
]
�I
)
⊂. , x n ]
XR[X
n
�
�
We can
]
,
.
.
.
,
x
]
�I
)
a
space
of
symmetric
matrices:
I
,
.
.
G G ∗
n
�
�
�
�
G
⊥
G
∗
�∗ = (R[x� , . . . , x n ]��IG )∗ with a space of symmetric matrices: IG
We
can
identify
R[X
]
⊥ R[x
∗
n ]]
� ,, .. .. .. ,, x
G
�
G
We
can
identify
R[X
]
=
(R[x
,
.
.
.
,
x
]
�I
)
with
a
space
of
symmetric
matrices:
I
R[x
x
R[x
,
.
.
.
,
x
]
.
n
n
�
� ℓG ∈ Σ X � PX ?
�
Grank
� is the nsmallest
What
ray
� of an extreme
G
�
G
G
What
�
P
XG ??
G
ΣX
�
P
What is
is the
the smallest
smallest rank
rank of
of an
an extreme
extreme ray
ray ℓℓ ∈∈ Σ
XG
XG
Theorem (Blekherman-Sinn-Velasco). Suppose m is the Green-Lazarsfeld index of IG , i.e. the
Theorem
(Blekherman-Sinn-Velasco).
Suppose
m
is
the
Green-Lazarsfeld
index
of
IIG ,, largest
i.e.
the
Theorem
(Blekherman-Sinn-Velasco).
Suppose
m
is
the
Green-Lazarsfeld
index
of
i.e.
the
(Blekherman-Sinn-Velasco).
Let
m
be
the
Green-Lazarsfeld
index
of
I
,
i.e.
the
largest integer p such that IG has property N�,p (the minimal free resolution is linear
G forGp steps).
largest
integer
pp such
that
IIG has
property
N
minimal
free
resolution
is
linear
for
pp steps).
�,p (the
largest
integer
such
that
has
property
N
(the
minimal
free
resolution
is
linear
for
steps).
integer
p
such
that
I
has
property
N
(the
minimal
free
resolution
is
linear
for
p
steps).
The
smallThe smallest rank ofGan extreme
∈ Σ XG ��,p
PXG is m + �. In terms of the graph, it is the smallest
G ray ℓ�,p
The
smallest
rank
of anray
extreme
ray �ℓ P
∈∈ Σ
�
P
m
+
��.. of
In terms
of theit graph,
it is the smallest
Xis
X�G. is
G
est
rank
of
an
extreme
ℓ
∈
Σ
m
+
In
terms
the
graph,
is
the
smallest
number
The
smallest
rank
of
an
extreme
ray
ℓ
Σ
�
P
is
m
+
In
terms
of
the
graph,
it is the smallest
number m ≥ � such that G contains
cycle
XG an induced
XGXG
XG of length m + �.
number m ≥ � such that G contains an induced cycle of length m + �.
m ≥ � such
G contains
an induced
cycle of length
+ �. m + �.
number
m ≥that
� such
that G contains
an induced
cycle ofmlength
Theorem (Eisenbud-Green-Hulek-Popescu). The Green-Lazarsfeld index of IG is the largest p
Theorem (Eisenbud-Green-Hulek-Popescu). The Green-Lazarsfeld index of IG is the largest p
Theorem
index of IG is the largest p
such that G(Eisenbud-Green-Hulek-Popescu).
has no induced cycle of length at mostThe
p +Green-Lazarsfeld
�.
such that G has no induced cycle of length at most p + �.
such that G has no induced cycle of length at most p + �.
References
References
References
Blekherman, Sinn, Velasco. Small Schemes and Sums of Squares, to appear on arXiv soon.
Blekherman, Sinn, Velasco. Small Schemes and Sums of Squares, to appear on arXiv soon.
Blekherman,
Velasco. Small
Schemes
and Sums
of Squares,
to appear
arXiv
soon.
Agler, Helton,Sinn,
McCullough,
Rodman.
Positive
semidefinite
matrices
with aon
given
sparsity
patAgler, Helton, McCullough, Rodman. Positive semidefinite matrices with a given sparsity patAgler,
Helton, McCullough,
Rodman.
Positive
semidefinite matrices
with a given
sparsity
pattern, Proceedings
of the Victoria
Conference
on Combinatorial
Matrix Analysis
(Victoria,
BC, ����)
tern, Proceedings of the Victoria Conference on Combinatorial Matrix Analysis (Victoria, BC, ����)
tern,
(����)Proceedings of the Victoria Conference on Combinatorial Matrix Analysis (Victoria, BC, ����)
(����)
The smallest rank of an extreme ray ℓ ∈ Σ XG � PXG is m + �. In terms of the graph, it is the smallest
number m ≥ � such that G contains an induced cycle of length m + �.
Theorem (Eisenbud-Green-Hulek-Popescu). The Green-Lazarsfeld index of IG is the largest p
such that G has no induced cycle of length at most p + �.
References
Blekherman, Sinn, Velasco. Small Schemes and Sums of Squares, to appear on arXiv soon.
Agler, Helton, McCullough, Rodman. Positive semidefinite matrices with a given sparsity pattern, Proceedings of the Victoria Conference on Combinatorial Matrix Analysis (Victoria, BC, ����)
(����)
Eisenbud, Green, Hulek, Popescu. Restricting linear syzygies: algebra and geometry, Compositio Mathematica (����)
Fröberg.On Stanley-Reisner rings, Topics in algebra, Part � (Warsaw, ����) (����)
Paulsen, Power, Smith. Schur products and matrix completions, Journal of Functional Analysis
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