## Abstract

We derandomize Valiant’s (J ACM 62, Article 13, 2015) subquadratic-time algorithm for finding outlier correlations in binary data. This demonstrates that it is possible to perform a deterministic subquadratic-time similarity join of high dimensionality. Our derandomized algorithm gives deterministic subquadratic scaling essentially for the same parameter range as Valiant’s randomized algorithm, but the precise constants we save over quadratic scaling are more modest. Our main technical tool for derandomization is an explicit family of correlation amplifiers built via a family of zigzag-product expanders by Reingold et al. (Ann Math 155(1):157–187, 2002). We say that a function f: { - 1 , 1 } ^{d}→ { - 1 , 1 } ^{D} is a correlation amplifier with threshold 0 ≤ τ≤ 1 , error γ≥ 1 , and strength p an even positive integer if for all pairs of vectors x, y∈ { - 1 , 1 } ^{d} it holds that (i) | ⟨ x, y⟩ | < τd implies | ⟨ f(x) , f(y) ⟩ | ≤ (τγ) ^{p}D; and (ii) | ⟨ x, y⟩ | ≥ τd implies (⟨x,y⟩γd)pD≤⟨f(x),f(y)⟩≤(γ⟨x,y⟩d)pD.

Original language | English |
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Pages (from-to) | 3306-3337 |

Number of pages | 32 |

Journal | Algorithmica |

Volume | 82 |

Issue number | 11 |

Early online date | 1 Jan 2020 |

DOIs | |

Publication status | Published - Nov 2020 |

MoE publication type | A1 Journal article-refereed |

## Keywords

- Correlation
- Derandomization
- Expander graph
- Outlier
- Similarity search