Two Universality Theorems
ShmuelOnn
Technion –Israel Institute of Technology
http://ie.technion.ac.il/~onn
Supported in part by ISF –Israel Science Foundation
Part 1:
GröbnerPolyhedra,Hilbert Polyhedra
and Universal GröbnerBases
Part 2:
MultiwayTables, Markov Bases
and ToricIdeals
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Part 1:
GröbnerPolyhedra, Hilbert Polyhedra
and Universal GröbnerBases
Based on joint papers with Sturmfelsand with Babson& Thomas
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GröbnerPolyhedra
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GröbnerPolyhedra
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Our Goal: Characterizationand efficient computation of
Gröbnerpolyhedraof classesofideals simultaneously.
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GröbnerPolyhedraof Point Configurations
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GröbnerPolyhedraof Point Configurations
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Hilbert Polyhedra
w
w
1
1
w
w
5
5
w
w
4
4
w
w3
3
G
w
w
2
2
w
w6
6
w
w
5
5
w
w4
4
w
w
6
6
w
w
2
2
w
w3
3
w
w
1
1
H
To handle Gröbnerpolyhedraof classes of ideals
simultaneously, we need the following universal object:
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Hilbert Polyhedra
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The Hilbert Scheme
Some examples of ideals on the Hilbert scheme are:
Unfortunately, this is a very complex object and may be intractable.
But, we next show how to approximate it efficiently and satisfactorily.
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The Hilbert Zonotope
Recall:
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Corollary:
Fix any d. The following hold for the Hilbert scheme:
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Some Problems:
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Part 2:
MultiwayTables, Markov Bases
and ToricIdeals
Based on several joint papers with Jesus De Loera
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MultiwayTables and Margins
0 2 2
2 1 0
4
3
2
3
2
Example:2table of size2
X 3with linesums:
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24
MultiwayTables and Margins
Example:3table of size3
X 4
X 6with a planesum:
0
3
5
0
3
3
2
0
1
4
1
2
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MultiwayTables and Margins
0
3
5
0
3
3
2
0
1
4
1
2
8
Example:3table of size3
X 4
X 6with a linesum:
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We consider two problemson the
set of all tableswith some margins fixed.
Mostly, we assume all linesums are fixed.
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Agencies such as the census bureau and center for health statistics
allow public webaccess to information on their data bases,
but areconcerned about confidentiality of individuals.
Common strategy:release marginsbut not table entries.
Question:how does theset of values that can occur in a
specific entryin all tables with the released marginslook like ?
Common perception:if the entryrangecontainsmany values
then theentryis secure; otherwise it is vulnerable.
Problem 1 Table Security: EntryRange
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0 2 2
2 1
0
4
3
2
3
2
2 2 0
0 1
2
4
3
2
3
2
1 2 1
1 1
1
4
3
2
3
2
Therefore, also the value 1occurs in that entry:
Specific example:the values 0, 2occur in an entry:
Example:for 2tables with fixed linesums,
theentryrangeis always aninterval.
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Theorem 1:
Theorem 1:For everyfinite set Sof nonnegative integers,
there are r,
c
and linesumsfor tables of size r
X
c
X
3
such that the set of values occurring in a fixed entryin all
possible tables with these linesumsis preciselyS.
In contrast, we show the following universalitytheoerm.
Example:for 2tables with fixed linesums,
theentryrangeis always aninterval.
Same holds for dtableswith fixedhyperplanesums.
So common practice is to compute by linear programming
lower bound
L
Land upper bound
U
Uon the possible values of
an entry and use the gap ULas a measure of its security.
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The only valuesoccurring in that entryin all
possible tables with these linesumsare 0,2
Example:Entryrangewith a gap
3
2
2
1
2
1
2
2
1
2
2
2
1
1
1
1
0
0
0
0
0
0
0
0
0
2
2
2
2
2
2
2
2
2
3
3
0
0
0
0
0
0
0
0
2
2
2
2
2
2
1
1
1
1
Consider the following linesumsfor tables of size 6
X
4
X
3:
Consider the
designated entry:
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Problem 2 Table Sampling: Markov Bases
AMarkov basisis a set of arraysthat enables a walk between
any two tableswith thesame marginswhile staying nonnegative.
It enables samplingthe (huge) set of tableswith fixed margins.
0 0 0
0 1 1
0 1 1
0 0 0
1 0 1
1 0 1
1 0 1
1 0 1
0 0 0
1 1 0
1 1 0
0 0 0
. . .
So Markov bases of rxc
tables with fixedlinesums aresimple:
they have constant support4and constant degree1regardless of r,c.
Same holds for dtableswith fixedhyperplanesums.
Example:Markov bases of rxc
tables with fixedlinesums are2x2minors:
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3tables with fixed linesums
Markov bases of 3tables with fixedlinesums aremuch more complicated.
In contrast, we show the following universalityof tables of size
rXcX3,
with one side3 fixedand smallest possibleandtwo sides
r,c
variable.
Theorem 2:
Theorem 2:For everyfinite set Vof integer vectors, there
are r,
c
such that any Markov basisfor r
X c
X 3tables with
fixed linesums,restricted to some entries, containsV.
So these Markov bases have unbounded degree and support.
Nice result(AokiTakemura, SantosSturmfels):for tables of size
rXcXh
,
with two sides
c,h
fixedandone side
r
variable,there is an upper bound
u(c,h) on degreeandsupportof Markov base elements,regardless of
r.
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Toricideals and Tables
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We have the following universalitytheorem for toricideals.
The equationsforcing the same marginson tables, such as linesums,
planesums, and so on, lift to a corresponding toricidealgenerated
by allbinomialscoming from pairs of tableswith the same margins:
Fundamental result (DiaconisSturmfels):thebinomials
xuxv
generate a
toricidealif and only ifthe corresponding arraysuvform a Markov basis.
Theorem 3:
Theorem 3:For everytoricideal I,there are
r,
c
such that any
generating set of the ideal of
r
X
c
X 3tables with fixed linesums,
restricted to some variables,contains a generating set ofI.
Toricideals and Tables
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Proofs: Universality of LineSum Polytopes
Theorems 13 stated before are corollaries
of the following remarkable theorem:
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Example Proof of Theorem 2:
Recall Theorem 2:
Recall Theorem 2:For everyfinite set Vof integer vectors,
there are r,
c
such that any Markov basisfor r
X c
X 3tables
with fixed linesums,restricted to some entries, containsV.
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Some Problems:
Classify Markov basesof hierarchical modelsdefined by table size
n1
x
. . .
x nd
and by simplicialcomplex of supportsof fixed margins:
Implications of the universality theoremon the existenceof a
strongly polynomial timealgorithm for linear programming?
Implications on the rational version of Hilbert’s 10th
problemon
the decidabilityof the realization problem for polytopes?
moderate, boundedmodels such as linesumsof r
x c
x
h
tables:
universal, intractable models such as linesumsof r
x c
x 3tables:
othermodels:
simple, squarefreemodels such as hyperplanesumsof dtables:
n1
. . .
n2
nd
r
c
h
3
c
r
n1
n
2
n
3
n
4
n
5
n
6
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