Fast Deterministic Algorithmsfor Matrix Completion Problems

Tasuku Soma

Research Institute for Mathematical Sciences,Kyoto Univ.

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 1 / 29

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-SymmetricMatrices

6 Conclusion

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 2 / 29

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-SymmetricMatrices

6 Conclusion

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 3 / 29

Matrix Completion

Matrix CompletionF: Field

Input Matrix A(x1, . . . , xn) over F(x1, . . . , xn) with indeterminatesx1, . . . , xn

Find α1, . . . , αn ∈ F maximizing rank A(α1, . . . , αn).

ExampleF = Q,

A =

[1 + x1 2 + x2

x3 x4

]−→ A ′ =

[2 21 0

](x1 := 1, x2 := 0, x3 := 1, x4 := 0)

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 4 / 29

Matrix Completion

Matrix CompletionF: Field

Input Matrix A(x1, . . . , xn) over F(x1, . . . , xn) with indeterminatesx1, . . . , xn

Find α1, . . . , αn ∈ F maximizing rank A(α1, . . . , αn).

ExampleF = Q,

A =

[1 + x1 2 + x2

x3 x4

]−→ A ′ =

[2 21 0

](x1 := 1, x2 := 0, x3 := 1, x4 := 0)

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 4 / 29

Backgrounds

A variety of combinatorial optimization problems can be formulated bymatrices with indeterminates:

Maximum matching,

Structural rigidity,

Network coding, etc.

Previous WorksMatrix completion for general matrices is solvable in polynomial timeby 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

Backgrounds

A variety of combinatorial optimization problems can be formulated bymatrices with indeterminates:

Maximum matching,

Structural rigidity,

Network coding, etc.

Previous WorksMatrix completion for general matrices is solvable in polynomial timeby 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 Results

Our 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 byrank-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

Our Results

Our 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 byrank-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

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-SymmetricMatrices

6 Conclusion

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 7 / 29

Problem Definition

Matrix Completion by Rank-One MatricesMatrix completion for A = B0 + x1B1 + · · ·+ xnBn, where B1, . . . ,Bn are ofrank one.

Example

B0 =

[1 00 0

], B1 =

[1 10 0

], B2 =

[2 01 0

]A =

[1 + x1 + 2x2 x1

x2 0

]

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 8 / 29

Problem Definition

Matrix Completion by Rank-One MatricesMatrix completion for A = B0 + x1B1 + · · ·+ xnBn, where B1, . . . ,Bn are ofrank one.

Example

B0 =

[1 00 0

], B1 =

[1 10 0

], B2 =

[2 01 0

]A =

[1 + x1 + 2x2 x1

x2 0

]

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 8 / 29

Previous Works

In 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 ’96

For the general case:

Ivanyos, Karpinski & Saxena ’10

An optimal solution can be found in O(m4.37n) 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

Previous Works

In 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 ’96

For the general case:

Ivanyos, Karpinski & Saxena ’10

An optimal solution can be found in O(m4.37n) 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

Previous Works

In 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 ’96

For the general case:

Ivanyos, Karpinski & Saxena ’10

An optimal solution can be found in O(m4.37n) 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

Idea

For A = B0 + x1B1 + · · ·+ xnBn (Bi = uiv>i (i = 1, . . . , n))

A :=

1. . .

10

v>1...

v>nx1

. . .

xn

1. . .

10

0 u1 · · · un B0

.

Lemma

rank A = 2n + rank A

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 10 / 29

Idea

For A = B0 + x1B1 + · · ·+ xnBn (Bi = uiv>i (i = 1, . . . , n))

A :=

1. . .

10

v>1...

v>nx1

. . .

xn

1. . .

10

0 u1 · · · un B0

.

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)

Harvey, Karger & Murota ’05

Matrix completion for a mixed matrix can be done in O(m2.77) time.

↓ Apply to A

TheoremMatrix 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

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

TheoremMatrix 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

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

TheoremMatrix 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 Theorem

TheoremFor A = B0 + x1B1 + · · ·+ xnBn,

max{rank A : x1, . . . , xn}

=min{

rank[

0 [vj : j < J]>

[uj : j ∈ J] B0

]: J ⊆ {1, . . . , n}

}.

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

Min-Max Theorem

TheoremFor A = B0 + x1B1 + · · ·+ xnBn,

max{rank A : x1, . . . , xn}

=min{

rank[

0 [vj : j < J]>

[uj : j ∈ J] B0

]: J ⊆ {1, . . . , n}

}.

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 Matrices

Simultaneous Matrix Completion by Rank-One MatricesF: Field

Input CollectionA 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

TheoremA 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

Simultaneous Matrix Completion by Rank-One Matrices

Simultaneous Matrix Completion by Rank-One MatricesF: Field

Input CollectionA 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

TheoremA 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 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-SymmetricMatrices

6 Conclusion

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 14 / 29

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

Multicast Problem with Linearly Correlated Sources

Messages in source nodes are linearlycorrelated

Each sink node demands the originalmessages x1 & x2

TheoremA 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 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-SymmetricMatrices

6 Conclusion

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 17 / 29

Problem Definition

Mixed Skew-Symmetric Matrix CompletionMatrix completion for a skew-symmetric matrix s.t. each indeterminateappears twice (mixed skew-symmetric matrix).

Example

A =

0 −1 11 0 0−1 0 0

+ 0 x 0−x 0 y0 −y 0

= 0 −1 + x 11 − x 0 y−1 −y 0

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 18 / 29

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 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

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 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 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 4 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 Pfaffian

Support graph:

A =

0 −2 1 12 0 0 3−1 0 0 21 −3 −2 0

Pfaffian:

pf A :=∑

M:perfect matching in G

±∏ij∈M

Aij

= A12A34 − A13A24

Lemmadet A = (pf A)2

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 21 / 29

Support Graph and Pfaffian

Support graph:

A =

0 −2 1 12 0 0 3−1 0 0 21 −3 −2 0

Pfaffian:

pf A :=∑

M:perfect matching in G

±∏ij∈M

Aij

= A12A34 − A13A24

Lemmadet A = (pf A)2

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 21 / 29

Sketch of Algorithm

Algorithm1: 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 do4: Substitute α to Tij so that Q[FQ ∪ {i, j}] will be nonsingular after

substitution.5: FQ := FQ ∪ {i, j}6: end for7: Substitute 0 to the rest of indeterminates8: return the resulting matrix

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 22 / 29

Sketch of Algorithm

Algorithm1: 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 do4: 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 indeterminates8: return the resulting matrix

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 22 / 29

Sketch of Algorithm

Algorithm1: 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 do4: Substitute α to Tij so that Q[FQ ∪ {i, j}] will be nonsingular after

substitution.5: FQ := FQ ∪ {i, j}6: end for7: Substitute 0 to the rest of indeterminates8: return the resulting matrix

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 22 / 29

Sketch of Algorithm

How can we find α s.t. Q[FQ ∪ {i, j}] will be nonsingular?

A = Q + T

A ′ = Q ′ + T ′

LemmaQ ′: 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

Sketch of Algorithm

How can we find α s.t. Q[FQ ∪ {i, j}] will be nonsingular?

A = Q + T A ′ = Q ′ + T ′

LemmaQ ′: 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

Sketch of Algorithm

How can we find α s.t. Q[FQ ∪ {i, j}] will be nonsingular?

A = Q + T A ′ = Q ′ + T ′

LemmaQ ′: 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

Sketch of Algorithm

Finally, 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 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-SymmetricMatrices

6 Conclusion

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 25 / 29

Problem Definition

Skew-Symmetric Matrix Completion by Rank-Two Skew-SymmetricMatricesMatrix 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

Our Result

In the case of B0 = 0:

Lovasz ’89This can be reduced to linear matroid parity.

solvable in O(m3n) 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

Our Result

In the case of B0 = 0:

Lovasz ’89This can be reduced to linear matroid parity.

solvable in O(m3n) 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 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-SymmetricMatrices

6 Conclusion

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 28 / 29

Conclusion

Our ResultsFaster algorithm and Min-Max theorem for matrix completion byrank-one matrices.

Application for multicast problem with linearly correlated sources.

First deterministic polynomial time algorithm for mixedskew-symmetric matrix completion.

First deterministic polynomial time algorithm for skew-symmetricmatrix completion by rank-two skew-symmetric matrices.

Future WorksApplication of skew-symmetric matrix completion

Matrix completion for other types of matrices

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 29 / 29

Conclusion

Our ResultsFaster algorithm and Min-Max theorem for matrix completion byrank-one matrices.

Application for multicast problem with linearly correlated sources.

First deterministic polynomial time algorithm for mixedskew-symmetric matrix completion.

First deterministic polynomial time algorithm for skew-symmetricmatrix completion by rank-two skew-symmetric matrices.

Future WorksApplication of skew-symmetric matrix completion

Matrix completion for other types of matrices

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 29 / 29