Riemannian gossip algorithms for decentralized matrix completion
The Complexity of Matrix Completion
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The Complexityof Matrix Completion
Nick HarveyDavid KargerSergey Yekhanin
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What is matrix completion?
Given matrix containing variables, substitute values for the variables to get full rank
1 x
1 y
1 1
1 0
x=1, y=0
1 x
1 y
1 1
1 1
x=1, y=1Bad
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Why should I care? Combinatorics
Many combinatorial problems relate to matrices of variables
Tutte ’47, Edmonds ’67, Lovasz ’79
Relation to Algebra
Tomizawa-Iri ’74, Murota ’00
Gessel-Viennot ’85
Graph Matching
Matroid Intersection
Counting paths in DAG
Problem
God(i.e., the BOOK)
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Why should I care? Algorithms
Often yields highly efficient algorithms
RNC: KUW’86, MVV’87Sequential O(n2.38) time: MS’04, H’06
O(nr1.38) time: H’06
Random Network Codes:Koetter-Medard ’03,
Ho et al. ’03
Graph Matching
Matroid Intersection
Counting paths in DAG
AlgorithmsProblem
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Why should I care? Complexity
Depending on parameters, can beNP-complete, in RP, or in PKey parameters:
Field size, # variables,# occurrences of each variable
Contains polynomial identity testing as special case (Valiant ’79)Derandomizing PIT implies strong circuit lower
bounds (Kabanets-Impagliazzo ’03)
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Field Size
Why care about field size?Relevant to complexity:
random works over large fieldsUnderstanding smaller fields may provide
insight to derandomization Important for network coding efficiency
(i.e., complexity of routers)
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Complexity Regions
Field Size
# Occurencesof an variable
2 3 5 7 n+1
1
2
3
4
5
6
7
8
9
22
NPHard
RP
P
Buss et al. ‘99Lovasz ‘79
H., Karger,Murota ‘05
P
Geelen ‘99? ? ? ? ? ?
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Complexity Regions
# Occurencesof an variable
2 3 5 7 n+1
1
2
3
4
5
6
7
8
9
22
NPHard
RP
NPHard
Field Size
P P
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Variant:Simultaneous Completion We have set of matrices A := {A1, …, Ad}
Each variable appears at most once per matrixAn variable can appear in several matrices
Def: A simultaneous completion for A assigns values to variables whilepreserving the rank of all matrices
RP algorithm still works over large field Application to Network Coding uses
Simultaneous Completion
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Relationship to Single Matrix Completion Hardness for Simultaneous Completion
Hardness for Single Matrix Completion w/many occurrences of variables
1 AB C
Simultaneous Completion
1 AD E
1 BC D
Single Matrix Completion
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Simultaneous Completion Algorithm
Input: d matrices
Compute rank of all matricesPick an variable x
for i {0,…,d}Set x := iIf all matrices have unchanged rank Recurse (# variables has decreased)
Simple self-reducibility algorithm Operates over field Fq, where d := # matrices < q
Non-trivial!Murota ’93.
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A Sharp Threshold Simple self-reducibility algorithm Operates over field Fq, where d := # matrices < q
Thm: Simultaneous completion for dmatrices over Fq is:
in P if q > d [HKM ’05] NP-hard if q ≤ d [This paper]
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A Sharp ThresholdThm: Simultaneous completion for d
matrices over Fq is:
in P if q > d [HKM ’05] NP-hard if q ≤ d [This paper]
Cor: Single matrix completion with d occurrences of variables over Fq
is NP-hard if q ≤ d
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Approach Reduction from Circuit-SAT
ANAND
BC
C = ( A B )
C = 1 - A ∙ B (if A, B, C {0, 1})
det 01 A
B C
(if A, B, C {0, 1})
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What have we shown so far? Simultaneous completion of an unbounded
number of matrices over F2 is NP-hard
Can we use fewer?Combine small matrices into huge matrix?Problem: Variables appear too many timesNeed to somehow make “copies” of a variable
Coming up next: completing two matrices over F2 is NP-hard
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A Curious Matrix
1 1 1 1 0
x1 1 1 1 1
x2 1 1 1
x3 1 1
xn 1
Rn :=
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A Curious Matrix
1 1 1 1 0
x1 1 1 1 1
x2 1 1 1
x3 1 1
xn 1
Rn :=
Thm: det Rn = i
ii
i xx )()1(
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Linearity of Determinant1 1 1 1 0
x1 1 1 1 1
x2 1 1 1
x3 1 1
xn 1
det
1 1 1 1 1
x1 1 1 1 1
x2 1 1 1
x3 1 1
xn 1
det
1 1 1 1 -1
x1 1 1 1 0
x2 1 1 0
x3 1 0
xn 0
det+=
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Column Expansion
ii
n x1)1(
x1 1 1 1
x2 1 1
x3 1
xn
(-1)n+1 det= =
1 1 1 1 1
x1 1 1 1 1
x2 1 1 1
x3 1 1
xn 1
det
1 1 1 1 -1
x1 1 1 1 0
x2 1 1 0
x3 1 0
xn 0
det+
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ii
n x1)1(
1 1 1 1 1
x1 1 1 1 1
x2 1 1 1
x3 1 1
xn 1
det
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Schur Complement Identity1 1 1 1 1
x1 1 1 1 1
x2 1 1 1
x3 1 1
xn 1
det
= det
x1 1 1 1
x2 1 1
x3 1
xn
1 1 1 11
1
1
1
1
∙ ∙ -
ii
n x1)1(
ii
n x1)1(
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Applying Outer Product
= det
1-x1
1 1-x2
1 1 1-x3
1 1 1 1-xn
= det
x1 1 1 1
x2 1 1
x3 1
xn
1 1 1 11
1
1
1
1
∙ ∙ -
ii
n x1)1(
ii
n x1)1(
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Finishing up
= det
1-x1
1 1-x2
1 1 1-x3
1 1 1 1-xn
i
ii
i xx )()1(=
QED
ii
n x1)1(
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Proof:
det Rn = ,
which is arithmetization of
So either all variables true, or all false.
Replicating VariablesCorollary:
If {x1, x2, …, xn} in {0,1}
then det Rn 0 xi = xj i,j
i
ii
i xx )()1(
xi xi. i i
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Replicating VariablesCorollary:
If {x1, x2, …, xn} in {0,1}
then det Rn 0 xi = xj i,j
Consequence: over F2, need only 2 matrices
NAND
NAND
NAND
A :=
Rn
Rn
Rn
B :=
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What have we shown so far?
Simultaneous completion of: an unbounded number of matrices
over F2 is NP-hard
two matrices over F2 is NP-hard
Next: q matrices over Fq is NP-hard
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Handling Fields Fq
Previous gadgets only work if each x {0,1}.How can we ensure this over Fq?
Introduce q-2 auxiliary variables: x=x(1), x(2), …, x(q-1)
Sufficient to enforce that:x(i) x(j) i,j and x(i) {0,1} i 2
det 01 1
x(i) x(j) etc.
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Handling Fields Fq
x(i) x(j) i,j and x(i) {0,1} i 2
x(2)
0 1
x(1)
x(3)x(4)
x(q-1)
Edge indicates endpoints non-equal
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Handling Fields Fq
x(i) x(j) i,j and x(i) {0,1} i 2
x(2)
0 1
x(1)
x(3)x(4)
x(q-1)
Pack these constraints into few matrices
Each variable used once per matrix
Amounts to edge-coloring From (Kn), conclude that
q matrices suffice
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What have we shown so far? Simultaneous completion of:
an unbounded number of matricesover F2 is NP-hard
two matrices over F2 is NP-hard
q matrices over Fq is NP-hard
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Main ResultsThm: A simultaneous completion for d
matrices over Fq is NP-hard if q ≤ d
Cor: Completion of single matrix, variables appearing d timesis NP-hard if q ≤ d
Cor: Completion of skew-symmetric matrix, variables appearing d timesis NP-hard if q ≤ d
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Open Questions Improved hardess results / algorithms
for matrix completion? Lower bounds / hardness for field size in
network coding? More combinatorial uses of matrix
completion