### 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) |
---|---|

Pages (from-to) | 542-553 |

Number of pages | 12 |

Journal | Discrete and Computational Geometry |

Volume | 43 |

Issue number | 3 |

DOIs | |

State | Published - Apr 2010 |

### Fingerprint

### Keywords

- Dimension reduction
- Johnson-Lindenstrauss lemma

### ASJC Scopus subject areas

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

### Cite this

*Discrete and Computational Geometry*,

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

**The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite.** / Johnson, William B.; Naor, Assaf.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 43, no. 3, pp. 542-553. https://doi.org/10.1007/s00454-009-9193-z

}

TY - JOUR

T1 - The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite

AU - Johnson, William B.

AU - Naor, Assaf

PY - 2010/4

Y1 - 2010/4

N2 - 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 x1,...,xn∈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}xi-xj{double pipe}≤{double pipe}L(xi)-L(xj){double pipe}≤O(1){dot operator}{double pipe}xi-xj{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 En⊆Y whose Euclidean distortion is at least 2Ω(α(n)), where α is the inverse Ackermann function.

AB - 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 x1,...,xn∈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}xi-xj{double pipe}≤{double pipe}L(xi)-L(xj){double pipe}≤O(1){dot operator}{double pipe}xi-xj{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 En⊆Y whose Euclidean distortion is at least 2Ω(α(n)), where α is the inverse Ackermann function.

KW - Dimension reduction

KW - Johnson-Lindenstrauss lemma

UR - http://www.scopus.com/inward/record.url?scp=77950521029&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77950521029&partnerID=8YFLogxK

U2 - 10.1007/s00454-009-9193-z

DO - 10.1007/s00454-009-9193-z

M3 - Article

VL - 43

SP - 542

EP - 553

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 3

ER -