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# Fast Deterministic Algorithms for Matrix Completion Problems

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### Fast Deterministic Algorithms for Matrix Completion Problems

1. 1. Fast Deterministic Algorithms for Matrix Completion Problems Tasuku Soma Research Institute for Mathematical Sciences, Kyoto Univ.Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 1 / 29
2. 2. 1 Introduction2 Matrix Completion by Rank-One Matrices3 Application to Network Coding4 Mixed Skew-Symmetric Matrix Completion5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric Matrices6 Conclusion Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 2 / 29
3. 3. 1 Introduction2 Matrix Completion by Rank-One Matrices3 Application to Network Coding4 Mixed Skew-Symmetric Matrix Completion5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric Matrices6 Conclusion Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 3 / 29
4. 4. Matrix CompletionMatrix CompletionF: Field Input Matrix A (x1 , . . . , xn ) over F(x1 , . . . , xn ) with indeterminates x1 , . . . , xn Find α1 , . . . , αn ∈ F maximizing rank A (α1 , . . . , αn ). Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 4 / 29
5. 5. Matrix CompletionMatrix CompletionF: Field Input Matrix A (x1 , . . . , xn ) over F(x1 , . . . , xn ) with indeterminates x1 , . . . , xn Find α1 , . . . , αn ∈ F maximizing rank A (α1 , . . . , αn ).ExampleF = Q, 1 + x1 2 + x2 2 2 A= −→ A = x3 x4 1 0 (x1 := 1, x2 := 0, x3 := 1, x4 := 0) Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 4 / 29
6. 6. BackgroundsA variety of combinatorial optimization problems can be formulated bymatrices with indeterminates: Maximum matching, Structural rigidity, Network coding, etc. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 5 / 29
7. 7. BackgroundsA variety of combinatorial optimization problems can be formulated bymatrices 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. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 5 / 29
8. 8. Our ResultsOur ResultsDeterministic polynomial time algorithms for the following matrixcompletion 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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 6 / 29
9. 9. Our ResultsOur ResultsDeterministic polynomial time algorithms for the following matrixcompletion 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 algorithmThey are working over an arbitrary ﬁeld! Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 6 / 29
10. 10. 1 Introduction2 Matrix Completion by Rank-One Matrices3 Application to Network Coding4 Mixed Skew-Symmetric Matrix Completion5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric Matrices6 Conclusion Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 7 / 29
11. 11. Problem DeﬁnitionMatrix Completion by Rank-One MatricesMatrix completion for A = B0 + x1 B1 + · · · + xn Bn , where B1 , . . . , Bn are ofrank one. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 8 / 29
12. 12. Problem DeﬁnitionMatrix Completion by Rank-One MatricesMatrix completion for A = B0 + x1 B1 + · · · + xn Bn , where B1 , . . . , Bn are ofrank one.Example 1 0 1 1 2 0B0 = , B1 = , B2 = 0 0 0 0 1 0 1 + x1 + 2x2 x1A= x2 0 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 8 / 29
13. 13. Previous WorksIn the case of B0 = 0: ´Lovasz ’89This can be reduced to linear matroid intersection. solvable in O (mn1.62 ) time using the algorithm of Gabow & Xu ’96m: the larger of row and column sizes, n: # of indeterminates Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 9 / 29
14. 14. Previous WorksIn the case of B0 = 0: ´Lovasz ’89This can be reduced to linear matroid intersection. solvable in O (mn1.62 ) time using the algorithm of Gabow & Xu ’96For the general case:Ivanyos, Karpinski & Saxena ’10An optimal solution can be found in O (m4.37 n) time.m: the larger of row and column sizes, n: # of indeterminates Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 9 / 29
15. 15. Previous WorksIn the case of B0 = 0: ´Lovasz ’89This can be reduced to linear matroid intersection. solvable in O (mn1.62 ) time using the algorithm of Gabow & Xu ’96For the general case:Ivanyos, Karpinski & Saxena ’10An optimal solution can be found in O (m4.37 n) time.Our ResultAn optimal solution can be found in O ((m + n)2.77 ) time.m: the larger of row and column sizes, n: # of indeterminates Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 9 / 29
16. 16. IdeaFor A = B0 + x1 B1 + · · · + xn Bn (Bi = ui vi (i = 1, . . . , n))  1 v1    .. .   0 .     . .                   1 vn       x1  1        ˜ :=  .. .. .   A 0   . .             xn 1                     0   B0   u1 ··· un         Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 10 / 29
17. 17. IdeaFor A = B0 + x1 B1 + · · · + xn Bn (Bi = ui vi (i = 1, . . . , n))  1 v1    .. .   0 .     . .                   1 vn       x1  1        ˜ :=  .. .. .   A 0   . .             xn 1                     0   B0   u1 ··· un        Lemma ˜rank A = 2n + rank A Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 10 / 29
18. 18. Algorithm ˜ ˜Each indeterminate appears only once in A ! (A is a mixed matrix) Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 11 / 29
19. 19. Algorithm ˜ ˜Each indeterminate appears only once in A ! (A is a mixed matrix)Harvey, Karger & Murota ’05Matrix completion for a mixed matrix can be done in O (m2.77 ) time.m: the larger of row and column sizes, n: # of indeterminates Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 11 / 29
20. 20. Algorithm ˜ ˜Each indeterminate appears only once in A ! (A is a mixed matrix)Harvey, Karger & Murota ’05Matrix completion for a mixed matrix can be done in O (m2.77 ) time. ˜ ↓ Apply to ATheoremMatrix 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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 11 / 29
21. 21. Min-Max TheoremTheoremFor A = B0 + x1 B1 + · · · + xn Bn , max{rank A : x1 , . . . , xn } 0 [vj : j J] = min rank : J ⊆ {1, . . . , n} . [uj : j ∈ J ] B0 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 12 / 29
22. 22. Min-Max TheoremTheoremFor A = B0 + x1 B1 + · · · + xn Bn , max{rank A : x1 , . . . , xn } 0 [vj : j J] = min rank : J ⊆ {1, . . . , n} . [uj : j ∈ J ] B0 ´Corollary (Lovasz ’89)If B0 = 0, then max{rank A : x1 , . . . , xn } = min{dim uj : j ∈ J + dim vj : j J : J ⊆ {1, . . . , n}} Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 12 / 29
23. 23. Simultaneous Matrix Completion by Rank-One MatricesSimultaneous Matrix Completion by Rank-One MatricesF: 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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 13 / 29
24. 24. Simultaneous Matrix Completion by Rank-One MatricesSimultaneous Matrix Completion by Rank-One MatricesF: 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 ATheoremA solution of simultaneous matrix completion by rank-one matrices can befound in polynomial time, if |F| > |A|. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 13 / 29
25. 25. 1 Introduction2 Matrix Completion by Rank-One Matrices3 Application to Network Coding4 Mixed Skew-Symmetric Matrix Completion5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric Matrices6 Conclusion Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 14 / 29
26. 26. Network CodingNetwork communication model s.t. intermediate nodes can perform coding Classical model Network coding Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 15 / 29
27. 27. Multicast Problem with Linearly Correlated Sources Messages in source nodes are linearly correlated Each sink node demands the original messages x1 & x2TheoremA solution of this multicast can be found in polynomial time.Approach: simultaneous matrix completion by rank-one matrices. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 16 / 29
28. 28. 1 Introduction2 Matrix Completion by Rank-One Matrices3 Application to Network Coding4 Mixed Skew-Symmetric Matrix Completion5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric Matrices6 Conclusion Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 17 / 29
29. 29. Problem DeﬁnitionMixed Skew-Symmetric Matrix CompletionMatrix completion for a skew-symmetric matrix s.t. each indeterminateappears twice (mixed skew-symmetric matrix).Example        0 −1 1  0       x 0  0     −1 + x 1 A =1 0 0 + −x 0 y  = 1 − x 0 y                          −1 0 0   0 −y 0   −1 −y  0 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 18 / 29
30. 30. Our ResultThere 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 inO (m4 ) time. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 19 / 29
31. 31. Our ResultThere 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 inO (m4 ) time.Our ResultMatrix completion for an m × m mixed skew-symmetric matrix can be donein O (m4 ) time. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 19 / 29
32. 32. Rank of Mixed Skew-Symmetric MatrixLemma (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). Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 20 / 29
33. 33. Support Graph and PfafﬁanSupport graph:  0 −2 1    1  2  0 0 3     A =    −1 0     0 2       1 −3 −2 0 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 21 / 29
34. 34. Support Graph and PfafﬁanSupport graph:  0 −2 1    1  2  0 0 3     A =    −1 0     0 2       1 −3 −2 0Pfafﬁan: pf A := ± Aij M :perfect matching in G ij ∈M = A12 A34 − A13 A24Lemma det A = (pf A )2 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 21 / 29
35. 35. Sketch of AlgorithmAlgorithm 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 ]. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 22 / 29
36. 36. Sketch of AlgorithmAlgorithm 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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 22 / 29
37. 37. Sketch of AlgorithmAlgorithm 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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 22 / 29
38. 38. Sketch of AlgorithmHow can we ﬁnd α s.t. Q [FQ ∪ {i , j }] will be nonsingular? A =Q +T Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 23 / 29
39. 39. Sketch of AlgorithmHow can we ﬁnd α s.t. Q [FQ ∪ {i , j }] will be nonsingular? A =Q +T A =Q +T Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 23 / 29
40. 40. Sketch of AlgorithmHow can we ﬁnd α s.t. Q [FQ ∪ {i , j }] will be nonsingular? A =Q +T A =Q +TLemmaQ : 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 ] Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 23 / 29
41. 41. Sketch of AlgorithmFinally, we obtain Q s.t. rank Q = rank A .TheoremMatrix completion for an m × m mixed skew-symmetric matrix can be donein O (m4 ) time. Using delta-covering algortihm of Geelen, Iwata & Murota ’03 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 24 / 29
42. 42. 1 Introduction2 Matrix Completion by Rank-One Matrices3 Application to Network Coding4 Mixed Skew-Symmetric Matrix Completion5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric Matrices6 Conclusion Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 25 / 29
43. 43. Problem DeﬁnitionSkew-Symmetric Matrix Completion by Rank-Two Skew-SymmetricMatricesMatrix completion for A = B0 + x1 B1 + · · · + xn Bn ,where B0 is skew-symmetric and B1 , . . . , Bn are rank-two skew-symmteric Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 26 / 29
44. 44. Our ResultIn the case of B0 = 0: ´Lovasz ’89This can be reduced to linear matroid parity. solvable in O (m3 n) time using the algorithm of Gabow & Stallman ’86. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 27 / 29
45. 45. Our ResultIn the case of B0 = 0: ´Lovasz ’89This 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 ResultAn 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) Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 27 / 29
46. 46. 1 Introduction2 Matrix Completion by Rank-One Matrices3 Application to Network Coding4 Mixed Skew-Symmetric Matrix Completion5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric Matrices6 Conclusion Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 28 / 29
47. 47. ConclusionOur 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. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 29 / 29
48. 48. ConclusionOur 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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 29 / 29