## Abstract

We design a sublinear-time approximation algorithm for quadratic function minimization problems with a better error bound than the previous algorithm by Hayashi and Yoshida (NIPS'16). Our approximation algorithm can be modified to handle the case where the minimization is done over a sphere. The analysis of our algorithms is obtained by combining results from graph limit theory, along with a novel spectral decomposition of matrices. Specifically, we prove that a matrix A can be decomposed into a structured part and a pseudorandom part, where the structured part is a block matrix with a polylogarithmic number of blocks, such that in each block all the entries are the same, and the pseudorandom part has a small spectral norm, achieving better error bound than the existing decomposition theorem of Frieze and Kannan (FOCS'96). As an additional application of the decomposition theorem, we give a sublinear-time approximation algorithm for computing the top singular values of a matrix.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018 |

Editors | Eric Blais, Jose D. P. Rolim, David Steurer, Klaus Jansen |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Print) | 9783959770859 |

DOIs | |

State | Published - 1 Aug 2018 |

Externally published | Yes |

Event | 21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018 - Princeton, United States Duration: 20 Aug 2018 → 22 Aug 2018 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 116 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018 |
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Country/Territory | United States |

City | Princeton |

Period | 20/08/18 → 22/08/18 |

### Bibliographical note

Publisher Copyright:© 2018 Aditya Bhaskara and Srivatsan Kumar.

## Keywords

- Approximation Algorithms
- Graph limits
- Matrix spectral decomposition
- Qudratic function minimization

## ASJC Scopus subject areas

- Software