### Abstract

Let X be a normed space that satisfies the Johnson-Lindenstrauss lemma (J-L lemma, in short) in the sense that for any integer n and any x_{1},...,x_{n}∈X, there exists a linear mapping L:X→F, where F⊆X is a linear subspace of dimension O(log n), such that {double pipe}x_{i}-x_{j}{double pipe}≤{double pipe}L(x_{i})-L(x_{j}){double pipe}≤O(1){dot operator}{double pipe}x_{i}-x_{j}{double pipe} for all i,j∈{1,...,n}. We show that this implies that X is almost Euclidean in the following sense: Every n-dimensional subspace of X embeds into Hilbert space with distortion,. On the other hand, we show that there exists a normed space Y which satisfies the J-L lemma, but for every n, there exists an n-dimensional subspace E_{n}⊆Y whose Euclidean distortion is at least 2^{Ω(α(n))}, where α is the inverse Ackermann function.

Original language | English (US) |
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Pages (from-to) | 542-553 |

Number of pages | 12 |

Journal | Discrete and Computational Geometry |

Volume | 43 |

Issue number | 3 |

DOIs | |

State | Published - Apr 1 2010 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Keywords

- Dimension reduction
- Johnson-Lindenstrauss lemma

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## Cite this

*Discrete and Computational Geometry*,

*43*(3), 542-553. https://doi.org/10.1007/s00454-009-9193-z