### 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 ∥x
_{i} - x
_{j}∥ ≤ ∥L(x
_{i}) - L(x
_{j})∥ ≤ O(1)·∥x
_{i} - x
_{j}∥ 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 2
^{2O(log * n)}. 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) |
---|---|

Title of host publication | Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms |

Pages | 885-891 |

Number of pages | 7 |

State | Published - 2009 |

Event | 20th Annual ACM-SIAM Symposium on Discrete Algorithms - New York, NY, United States Duration: Jan 4 2009 → Jan 6 2009 |

### Other

Other | 20th Annual ACM-SIAM Symposium on Discrete Algorithms |
---|---|

Country | United States |

City | New York, NY |

Period | 1/4/09 → 1/6/09 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms*(pp. 885-891)

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

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms.*pp. 885-891, 20th Annual ACM-SIAM Symposium on Discrete Algorithms, New York, NY, United States, 1/4/09.

}

TY - GEN

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

AU - Johnson, William B.

AU - Naor, Assaf

PY - 2009

Y1 - 2009

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 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 ∥x i - x j∥ ≤ ∥L(x i) - L(x j)∥ ≤ O(1)·∥x i - x j∥ 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 2 2O(log * n). 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.

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 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 ∥x i - x j∥ ≤ ∥L(x i) - L(x j)∥ ≤ O(1)·∥x i - x j∥ 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 2 2O(log * n). 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.

UR - http://www.scopus.com/inward/record.url?scp=70349102481&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70349102481&partnerID=8YFLogxK

M3 - Conference contribution

SN - 9780898716801

SP - 885

EP - 891

BT - Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms

ER -