Fast Deterministic Algorithms for Matrix Completion Problems
1. Fast Deterministic Algorithms
for Matrix Completion Problems
Tasuku Soma
Research Institute for Mathematical Sciences,
Kyoto Univ.
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2. 1 Introduction
2 Matrix Completion by Rank-One Matrices
3 Application to Network Coding
4 Mixed Skew-Symmetric Matrix Completion
5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric
Matrices
6 Conclusion
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3. 1 Introduction
2 Matrix Completion by Rank-One Matrices
3 Application to Network Coding
4 Mixed Skew-Symmetric Matrix Completion
5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric
Matrices
6 Conclusion
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4. Matrix Completion
Matrix Completion
F: Field
Input Matrix A (x1 , . . . , xn ) over F(x1 , . . . , xn ) with indeterminates
x1 , . . . , xn
Find α1 , . . . , αn ∈ F maximizing rank A (α1 , . . . , αn ).
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5. Matrix Completion
Matrix Completion
F: Field
Input Matrix A (x1 , . . . , xn ) over F(x1 , . . . , xn ) with indeterminates
x1 , . . . , xn
Find α1 , . . . , αn ∈ F maximizing rank A (α1 , . . . , αn ).
Example
F = Q,
1 + x1 2 + x2 2 2
A= −→ A =
x3 x4 1 0
(x1 := 1, x2 := 0, x3 := 1, x4 := 0)
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6. Backgrounds
A variety of combinatorial optimization problems can be formulated by
matrices with indeterminates:
Maximum matching,
Structural rigidity,
Network coding, etc.
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7. Backgrounds
A variety of combinatorial optimization problems can be formulated by
matrices with indeterminates:
Maximum matching,
Structural rigidity,
Network coding, etc.
Previous Works
Matrix completion for general matrices is solvable in polynomial time
by a randomized algorithm if the field is sufficiently large.
Deterministic algorithms are known only for special matrices
(cf. polynomial identity testing)
NP hard over a general field.
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8. Our Results
Our Results
Deterministic polynomial time algorithms for the following matrix
completion problems:
Matrix completion by rank-one matrices
— a faster algorithm than the previous one
Mixed skew-symmetric matrix completion
— the first deterministic polynomial time algorithm
Skew-symmetric matrix completion by
rank-two skew-symmetric matrices
— the first deterministic polynomial time algorithm
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9. Our Results
Our Results
Deterministic polynomial time algorithms for the following matrix
completion problems:
Matrix completion by rank-one matrices
— a faster algorithm than the previous one
Mixed skew-symmetric matrix completion
— the first deterministic polynomial time algorithm
Skew-symmetric matrix completion by
rank-two skew-symmetric matrices
— the first deterministic polynomial time algorithm
They are working over an arbitrary field!
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10. 1 Introduction
2 Matrix Completion by Rank-One Matrices
3 Application to Network Coding
4 Mixed Skew-Symmetric Matrix Completion
5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric
Matrices
6 Conclusion
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11. Problem Definition
Matrix Completion by Rank-One Matrices
Matrix completion for A = B0 + x1 B1 + · · · + xn Bn , where B1 , . . . , Bn are of
rank one.
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12. Problem Definition
Matrix Completion by Rank-One Matrices
Matrix completion for A = B0 + x1 B1 + · · · + xn Bn , where B1 , . . . , Bn are of
rank one.
Example
1 0 1 1 2 0
B0 = , B1 = , B2 =
0 0 0 0 1 0
1 + x1 + 2x2 x1
A=
x2 0
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13. Previous Works
In the case of B0 = 0:
´
Lovasz ’89
This can be reduced to linear matroid intersection.
solvable in O (mn1.62 ) time using the algorithm of Gabow & Xu ’96
m: the larger of row and column sizes, n: # of indeterminates
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14. Previous Works
In the case of B0 = 0:
´
Lovasz ’89
This can be reduced to linear matroid intersection.
solvable in O (mn1.62 ) time using the algorithm of Gabow & Xu ’96
For the general case:
Ivanyos, Karpinski & Saxena ’10
An optimal solution can be found in O (m4.37 n) time.
m: the larger of row and column sizes, n: # of indeterminates
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15. Previous Works
In the case of B0 = 0:
´
Lovasz ’89
This can be reduced to linear matroid intersection.
solvable in O (mn1.62 ) time using the algorithm of Gabow & Xu ’96
For the general case:
Ivanyos, Karpinski & Saxena ’10
An optimal solution can be found in O (m4.37 n) time.
Our Result
An optimal solution can be found in O ((m + n)2.77 ) time.
m: the larger of row and column sizes, n: # of indeterminates
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18. Algorithm
˜ ˜
Each indeterminate appears only once in A ! (A is a mixed matrix)
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19. Algorithm
˜ ˜
Each indeterminate appears only once in A ! (A is a mixed matrix)
Harvey, Karger & Murota ’05
Matrix completion for a mixed matrix can be done in O (m2.77 ) time.
m: the larger of row and column sizes, n: # of indeterminates
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20. Algorithm
˜ ˜
Each indeterminate appears only once in A ! (A is a mixed matrix)
Harvey, Karger & Murota ’05
Matrix completion for a mixed matrix can be done in O (m2.77 ) time.
˜
↓ Apply to A
Theorem
Matrix completion by rank-one matrices can be done in O ((m + n)2.77 )
time.
m: the larger of row and column sizes, n: # of indeterminates
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23. Simultaneous Matrix Completion by Rank-One Matrices
Simultaneous Matrix Completion by Rank-One Matrices
F: Field
Input Collection A of matrices in the form of B0 + x1 B1 + · · · + xn Bn
Find Value assignments αi ∈ F for each indeterminate xi
maximizing the rank of every matrix in A
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24. Simultaneous Matrix Completion by Rank-One Matrices
Simultaneous Matrix Completion by Rank-One Matrices
F: Field
Input Collection A of matrices in the form of B0 + x1 B1 + · · · + xn Bn
Find Value assignments αi ∈ F for each indeterminate xi
maximizing the rank of every matrix in A
Theorem
A solution of simultaneous matrix completion by rank-one matrices can be
found in polynomial time, if |F| > |A|.
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25. 1 Introduction
2 Matrix Completion by Rank-One Matrices
3 Application to Network Coding
4 Mixed Skew-Symmetric Matrix Completion
5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric
Matrices
6 Conclusion
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26. Network Coding
Network communication model s.t. intermediate nodes can perform coding
Classical model Network coding
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27. Multicast Problem with Linearly Correlated Sources
Messages in source nodes are linearly
correlated
Each sink node demands the original
messages x1 & x2
Theorem
A solution of this multicast can be found in polynomial time.
Approach: simultaneous matrix completion by rank-one matrices.
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28. 1 Introduction
2 Matrix Completion by Rank-One Matrices
3 Application to Network Coding
4 Mixed Skew-Symmetric Matrix Completion
5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric
Matrices
6 Conclusion
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30. Our Result
There were no algorithms for this problem, but we can compute the rank.
Murota ’03 (←Geelen, Iwata & Murota ’03)
The rank of an m × m mixed skew-symmetric matrix can be computed in
O (m4 ) time.
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31. Our Result
There were no algorithms for this problem, but we can compute the rank.
Murota ’03 (←Geelen, Iwata & Murota ’03)
The rank of an m × m mixed skew-symmetric matrix can be computed in
O (m4 ) time.
Our Result
Matrix completion for an m × m mixed skew-symmetric matrix can be done
in O (m4 ) time.
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32. Rank of Mixed Skew-Symmetric Matrix
Lemma (Murota ’03)
For an m × m mixed skew-symmetric matrix A = Q + T
(Q: constant part, T : indeterminates part),
rank A = max |FQ FT | : both Q [FQ ], T [FT ] are nonsingular
RHS is linear delta-covering.
Optimal FQ and FT can be found in O (m4 )
time (Geelen, Iwata & Murota ’03).
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34. Support Graph and Pfaffian
Support graph:
0 −2 1
1
2
0 0 3
A =
−1 0
0 2
1 −3 −2 0
Pfaffian:
pf A := ± Aij
M :perfect matching in G ij ∈M
= A12 A34 − A13 A24
Lemma
det A = (pf A )2
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35. Sketch of Algorithm
Algorithm
1: Find an optimal solution FQ and FT for linear delta-covering.
2: Find a perfect matching M in the support graph of T [FT ].
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36. Sketch of Algorithm
Algorithm
1: Find an optimal solution FQ and FT for linear delta-covering.
2: Find a perfect matching M in the support graph of T [FT ].
3: for each ij ∈ M do
4: Substitute α to Tij so that Q [FQ ∪ {i , j }] will be nonsingular after
substitution.
5: FQ := FQ ∪ {i , j }
6: end for
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37. Sketch of Algorithm
Algorithm
1: Find an optimal solution FQ and FT for linear delta-covering.
2: Find a perfect matching M in the support graph of T [FT ].
3: for each ij ∈ M do
4: Substitute α to Tij so that Q [FQ ∪ {i , j }] will be nonsingular after
substitution.
5: FQ := FQ ∪ {i , j }
6: end for
7: Substitute 0 to the rest of indeterminates
8: return the resulting matrix
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38. Sketch of Algorithm
How can we find α s.t. Q [FQ ∪ {i , j }] will be nonsingular?
A =Q +T
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39. Sketch of Algorithm
How can we find α s.t. Q [FQ ∪ {i , j }] will be nonsingular?
A =Q +T A =Q +T
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40. Sketch of Algorithm
How can we find α s.t. Q [FQ ∪ {i , j }] will be nonsingular?
A =Q +T A =Q +T
Lemma
Q : modified matrix of Q as Qij := Qij + α, Qji := Qji − α
pf Q [FQ ∪ {i , j }] = pf Q [FQ ∪ {i , j }] ± α · pf Q [FQ ]
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41. Sketch of Algorithm
Finally, we obtain Q s.t. rank Q = rank A .
Theorem
Matrix completion for an m × m mixed skew-symmetric matrix can be done
in O (m4 ) time.
Using delta-covering algortihm of Geelen, Iwata & Murota ’03
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42. 1 Introduction
2 Matrix Completion by Rank-One Matrices
3 Application to Network Coding
4 Mixed Skew-Symmetric Matrix Completion
5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric
Matrices
6 Conclusion
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43. Problem Definition
Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric
Matrices
Matrix completion for A = B0 + x1 B1 + · · · + xn Bn ,
where B0 is skew-symmetric and B1 , . . . , Bn are rank-two skew-symmteric
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44. Our Result
In the case of B0 = 0:
´
Lovasz ’89
This can be reduced to linear matroid parity.
solvable in O (m3 n) time using the algorithm of Gabow & Stallman ’86.
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45. Our Result
In the case of B0 = 0:
´
Lovasz ’89
This can be reduced to linear matroid parity.
solvable in O (m3 n) time using the algorithm of Gabow & Stallman ’86.
For the general case:
Our Result
An optimal solution can be found in O ((m + n)4 ) time.
Idea: Reduction to mixed skew-symmetric matrix completion
(similar to matrix completion by rank-one matrices)
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46. 1 Introduction
2 Matrix Completion by Rank-One Matrices
3 Application to Network Coding
4 Mixed Skew-Symmetric Matrix Completion
5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric
Matrices
6 Conclusion
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47. Conclusion
Our Results
Faster algorithm and Min-Max theorem for matrix completion by
rank-one matrices.
Application for multicast problem with linearly correlated sources.
First deterministic polynomial time algorithm for mixed
skew-symmetric matrix completion.
First deterministic polynomial time algorithm for skew-symmetric
matrix completion by rank-two skew-symmetric matrices.
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48. Conclusion
Our Results
Faster algorithm and Min-Max theorem for matrix completion by
rank-one matrices.
Application for multicast problem with linearly correlated sources.
First deterministic polynomial time algorithm for mixed
skew-symmetric matrix completion.
First deterministic polynomial time algorithm for skew-symmetric
matrix completion by rank-two skew-symmetric matrices.
Future Works
Application of skew-symmetric matrix completion
Matrix completion for other types of matrices
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