### Abstract

The little Grothendieck problem consists of maximizing ∑ _{i} _{j}C_{i} _{j}x_{i}x_{j} for a positive semidefinite matrix C, over binary variables x_{i}∈ { Â± 1 }. In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given C∈ R^{d} ^{n} ^{Ã—} ^{d} ^{n} a positive semidefinite matrix, the objective is to maximize ∑ijtr(CijTOiOjT) restricting O_{i} to take values in the group of orthogonal matrices O_{d}, where C_{i} _{j} denotes the (ij)-th dÃ— d block of C. We propose an approximation algorithm, which we refer to as Orthogonal-Cut, to solve the little Grothendieck problem over the group of orthogonal matrices O_{d} and show a constant approximation ratio. Our method is based on semidefinite programming. For a given d≥ 1 , we show a constant approximation ratio of α_{R}(d) ^{2}, where α_{R}(d) is the expected average singular value of a dÃ— d matrix with random Gaussian N(0,1d) i.i.d. entries. For d= 1 we recover the known α_{R}(1) ^{2}= 2 / π approximation guarantee for the classical little Grothendieck problem. Our algorithm and analysis naturally extends to the complex valued case also providing a constant approximation ratio for the analogous little Grothendieck problem over the Unitary Group U_{d}. Orthogonal-Cut also serves as an approximation algorithm for several applications, including the Procrustes problem where it improves over the best previously known approximation ratio ofÂ 122. The little Grothendieck problem falls under the larger class of problems approximated by a recent algorithm proposed in the context of the non-commutative Grothendieck inequality. Nonetheless, our approach is simpler and provides better approximation with matching integrality gaps. Finally, we also provide an improved approximation algorithm for the more general little Grothendieck problem over the orthogonal (or unitary) group with rank constraints, recovering, when d= 1 , the sharp, known ratios.

Original language | English (US) |
---|---|

Pages (from-to) | 433-475 |

Number of pages | 43 |

Journal | Mathematical Programming |

Volume | 160 |

Issue number | 1-2 |

DOIs | |

State | Published - Nov 1 2016 |

### Fingerprint

### Keywords

- Approximation algorithms
- Procrustes problem
- Semidefinite programming

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*Mathematical Programming*,

*160*(1-2), 433-475. https://doi.org/10.1007/s10107-016-0993-7

**Approximating the little Grothendieck problem over the orthogonal and unitary groups.** / Bandeira, Afonso; Kennedy, Christopher; Singer, Amit.

Research output: Contribution to journal › Article

*Mathematical Programming*, vol. 160, no. 1-2, pp. 433-475. https://doi.org/10.1007/s10107-016-0993-7

}

TY - JOUR

T1 - Approximating the little Grothendieck problem over the orthogonal and unitary groups

AU - Bandeira, Afonso

AU - Kennedy, Christopher

AU - Singer, Amit

PY - 2016/11/1

Y1 - 2016/11/1

N2 - The little Grothendieck problem consists of maximizing ∑ i jCi jxixj for a positive semidefinite matrix C, over binary variables xi∈ { Â± 1 }. In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given C∈ Rd n Ã— d n a positive semidefinite matrix, the objective is to maximize ∑ijtr(CijTOiOjT) restricting Oi to take values in the group of orthogonal matrices Od, where Ci j denotes the (ij)-th dÃ— d block of C. We propose an approximation algorithm, which we refer to as Orthogonal-Cut, to solve the little Grothendieck problem over the group of orthogonal matrices Od and show a constant approximation ratio. Our method is based on semidefinite programming. For a given d≥ 1 , we show a constant approximation ratio of αR(d) 2, where αR(d) is the expected average singular value of a dÃ— d matrix with random Gaussian N(0,1d) i.i.d. entries. For d= 1 we recover the known αR(1) 2= 2 / π approximation guarantee for the classical little Grothendieck problem. Our algorithm and analysis naturally extends to the complex valued case also providing a constant approximation ratio for the analogous little Grothendieck problem over the Unitary Group Ud. Orthogonal-Cut also serves as an approximation algorithm for several applications, including the Procrustes problem where it improves over the best previously known approximation ratio ofÂ 122. The little Grothendieck problem falls under the larger class of problems approximated by a recent algorithm proposed in the context of the non-commutative Grothendieck inequality. Nonetheless, our approach is simpler and provides better approximation with matching integrality gaps. Finally, we also provide an improved approximation algorithm for the more general little Grothendieck problem over the orthogonal (or unitary) group with rank constraints, recovering, when d= 1 , the sharp, known ratios.

AB - The little Grothendieck problem consists of maximizing ∑ i jCi jxixj for a positive semidefinite matrix C, over binary variables xi∈ { Â± 1 }. In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given C∈ Rd n Ã— d n a positive semidefinite matrix, the objective is to maximize ∑ijtr(CijTOiOjT) restricting Oi to take values in the group of orthogonal matrices Od, where Ci j denotes the (ij)-th dÃ— d block of C. We propose an approximation algorithm, which we refer to as Orthogonal-Cut, to solve the little Grothendieck problem over the group of orthogonal matrices Od and show a constant approximation ratio. Our method is based on semidefinite programming. For a given d≥ 1 , we show a constant approximation ratio of αR(d) 2, where αR(d) is the expected average singular value of a dÃ— d matrix with random Gaussian N(0,1d) i.i.d. entries. For d= 1 we recover the known αR(1) 2= 2 / π approximation guarantee for the classical little Grothendieck problem. Our algorithm and analysis naturally extends to the complex valued case also providing a constant approximation ratio for the analogous little Grothendieck problem over the Unitary Group Ud. Orthogonal-Cut also serves as an approximation algorithm for several applications, including the Procrustes problem where it improves over the best previously known approximation ratio ofÂ 122. The little Grothendieck problem falls under the larger class of problems approximated by a recent algorithm proposed in the context of the non-commutative Grothendieck inequality. Nonetheless, our approach is simpler and provides better approximation with matching integrality gaps. Finally, we also provide an improved approximation algorithm for the more general little Grothendieck problem over the orthogonal (or unitary) group with rank constraints, recovering, when d= 1 , the sharp, known ratios.

KW - Approximation algorithms

KW - Procrustes problem

KW - Semidefinite programming

UR - http://www.scopus.com/inward/record.url?scp=84961665160&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84961665160&partnerID=8YFLogxK

U2 - 10.1007/s10107-016-0993-7

DO - 10.1007/s10107-016-0993-7

M3 - Article

VL - 160

SP - 433

EP - 475

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1-2

ER -