Two Universality Theorems

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8 Οκτ 2013 (πριν από 3 χρόνια και 10 μήνες)

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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
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5
5
w
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4
4
w
w3
3
G
w
w
2
2
w
w6
6
w
w
5
5
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w4
4
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6
6
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2
2
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w3
3
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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:2-table of size2
X 3with line-sums:
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MultiwayTables and Margins
Example:3-table of size3
X 4
X 6with a plane-sum:
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:3-table of size3
X 4
X 6with a line-sum:
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We consider two problemson the
set of all tableswith some margins fixed.
Mostly, we assume all line-sums are fixed.
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Agencies such as the census bureau and center for health statistics
allow public web-access 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 entry-rangecontainsmany values
then theentryis secure; otherwise it is vulnerable.
Problem 1 -Table Security: Entry-Range
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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 2-tables with fixed line-sums,
theentry-rangeis always aninterval.
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Theorem 1:
Theorem 1:For everyfinite set Sof nonnegative integers,
there are r,
c
and line-sumsfor tables of size r
X
c
X
3
such that the set of values occurring in a fixed entryin all
possible tables with these line-sumsis preciselyS.
In contrast, we show the following universalitytheoerm.
Example:for 2-tables with fixed line-sums,
theentry-rangeis always aninterval.
Same holds for d-tableswith fixedhyperplane-sums.
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 U-Las a measure of its security.
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The only valuesoccurring in that entryin all
possible tables with these line-sumsare 0,2
Example:Entry-rangewith 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 line-sumsfor 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 fixedline-sums aresimple:
they have constant support4and constant degree1regardless of r,c.
Same holds for d-tableswith fixedhyperplane-sums.
Example:Markov bases of rxc
tables with fixedline-sums are2x2minors:
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3-tables with fixed line-sums
Markov bases of 3-tables with fixedline-sums 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 line-sums,restricted to some entries, containsV.
So these Markov bases have unbounded degree and support.
Nice result(Aoki-Takemura, Santos-Sturmfels):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 line-sums,
plane-sums, and so on, lift to a corresponding toricidealgenerated
by allbinomialscoming from pairs of tableswith the same margins:
Fundamental result (Diaconis-Sturmfels):thebinomials
xu-xv
generate a
toricidealif and only ifthe corresponding arraysu-vform 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 line-sums,
restricted to some variables,contains a generating set ofI.
Toricideals and Tables
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Proofs: Universality of Line-Sum Polytopes
Theorems 1-3 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 line-sums,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 line-sumsof r
x c
x
h
tables:
-universal, intractable models such as line-sumsof r
x c
x 3tables:
-othermodels:
-simple, square-freemodels such as hyperplane-sumsof d-tables:
n1
. . .
n2
nd
r
c
h
3
c
r
n1
n
2
n
3
n
4
n
5
n
6
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