Fast Deterministic Algorithms for Matrix Completion Problems

IPCO 2013

Transcript

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. 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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 2 / 29
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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 3 / 29
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). Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 4 / 29
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, A = 1 + x1 2 + x2 x3 x4 −→ A = 2 2 1 0 (x1 := 1, x2 := 0, x3 := 1, x4 := 0) Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 4 / 29
6. Backgrounds A variety of combinatorial optimization problems can be formulated by matrices with indeterminates: Maximum matching, Structural rigidity, Network coding, etc. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 5 / 29
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. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 5 / 29
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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 6 / 29
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! Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 6 / 29
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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 7 / 29
11. Problem Definition Matrix Completion by Rank-One Matrices Matrix completion for A = B0 + x1B1 + · · · + xnBn, where B1 , . . . , Bn are of rank one. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 8 / 29
12. Problem Definition Matrix Completion by Rank-One Matrices Matrix completion for A = B0 + x1B1 + · · · + xnBn, where B1 , . . . , Bn are of rank one. Example B0 = 1 0 0 0 , B1 = 1 1 0 0 , B2 = 2 0 1 0 A = 1 + x1 + 2x2 x1 x2 0 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 8 / 29
13. Previous Works In the case of B0 = 0: Lov´ asz ’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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 9 / 29
14. Previous Works In the case of B0 = 0: Lov´ asz ’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(m4.37n) time. m: the larger of row and column sizes, n: # of indeterminates Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 9 / 29
15. Previous Works In the case of B0 = 0: Lov´ asz ’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(m4.37n) 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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 9 / 29
16. Idea For A = B0 + x1B1 + · · · + xnBn (Bi = uiv i (i = 1, . . . , n)) ˜ A :=                                               1 ... 1 0 v 1 . . . vn x1 ... xn 1 ... 1 0 0 u1 · · · un B0                                               . Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 10 / 29
17. Idea For A = B0 + x1B1 + · · · + xnBn (Bi = uiv i (i = 1, . . . , n)) ˜ A :=                                               1 ... 1 0 v 1 . . . vn x1 ... xn 1 ... 1 0 0 u1 · · · un B0                                               . Lemma rank ˜ A = 2n + rank A Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 10 / 29
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. 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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 11 / 29
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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 11 / 29
21. Min-Max Theorem Theorem For A = B0 + x1B1 + · · · + xnBn, max{rank A : x1 , . . . , xn} = min rank 0 [vj : j J] [uj : j ∈ J] B0 : J ⊆ {1, . . . , n} . Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 12 / 29
22. Min-Max Theorem Theorem For A = B0 + x1B1 + · · · + xnBn, max{rank A : x1 , . . . , xn} = min rank 0 [vj : j J] [uj : j ∈ J] B0 : J ⊆ {1, . . . , n} . Corollary (Lov´ asz ’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. 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 + x1B1 + · · · + xnBn 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. 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 + x1B1 + · · · + xnBn 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|. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 13 / 29
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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 14 / 29
26. Network Coding Network communication model s.t. intermediate nodes can perform coding Classical model Network coding Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 15 / 29
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. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 16 / 29
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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 17 / 29
29. Problem Definition Mixed Skew-Symmetric Matrix Completion Matrix completion for a skew-symmetric matrix s.t. each indeterminate appears twice (mixed skew-symmetric matrix). Example A =           0 −1 1 1 0 0 −1 0 0           +           0 x 0 −x 0 y 0 −y 0           =           0 −1 + x 1 1 − x 0 y −1 −y 0           Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 18 / 29
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. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 19 / 29
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. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 19 / 29
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). Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 20 / 29
33. Support Graph and Pfaffian Support graph: A =                0 −2 1 1 2 0 0 3 −1 0 0 2 1 −3 −2 0                Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 21 / 29
34. Support Graph and Pfaffian Support graph: A =                0 −2 1 1 2 0 0 3 −1 0 0 2 1 −3 −2 0                Pfaffian: pf A := M:perfect matching in G ± ij∈M Aij = A12A34 − A13A24 Lemma det A = (pf A)2 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 21 / 29
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 ]. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 22 / 29
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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 22 / 29
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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 22 / 29
38. Sketch of Algorithm How 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. Sketch of Algorithm How 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. 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 Q ij := Qij + α, Q ji := 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. 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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 24 / 29
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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 25 / 29
43. Problem Definition Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric Matrices Matrix completion for A = B0 + x1B1 + · · · + xnBn, where B0 is skew-symmetric and B1 , . . . , Bn are rank-two skew-symmteric Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 26 / 29
44. Our Result In the case of B0 = 0: Lov´ asz ’89 This can be reduced to linear matroid parity. solvable in O(m3n) time using the algorithm of Gabow & Stallman ’86. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 27 / 29
45. Our Result In the case of B0 = 0: Lov´ asz ’89 This can be reduced to linear matroid parity. solvable in O(m3n) 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) Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 27 / 29
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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 28 / 29
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. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 29 / 29
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 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 29 / 29