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Fast Deterministic Algorithms for Matrix Completion Problems

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    Fast Deterministic Algorithms for Matrix Completion Problems Fast Deterministic Algorithms for Matrix Completion Problems Presentation Transcript

    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • Algorithm ˜ ˜Each indeterminate appears only once in A ! (A is a mixed matrix) Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 11 / 29
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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