Randomized large distortion dimension reduction

Randomized large distortion dimension reduction Consider a random matrix $$H:{\mathbb {R}}^{n}\longrightarrow {\mathbb {R}}^{m}$$ H : R n ⟶ R m . Let $$D\ge 2$$ D ≥ 2 and let $$\{W_l\}_{l=1}^{p}$$ { W l } l = 1 p be a set of $$k$$ k -dimensional affine subspaces of $${\mathbb {R}}^{n}$$ R n . We ask what is the probability that for all $$1\le l\le p$$ 1 ≤ l ≤ p and $$x,y\in W_l$$ x , y ∈ W l , $$\begin{aligned} \Vert x-y\Vert _2\le \Vert Hx-Hy\Vert _2\le D\Vert x-y\Vert _2. \end{aligned}$$ ‖ x - y ‖ 2 ≤ ‖ H x - H y ‖ 2 ≤ D ‖ x - y ‖ 2 . We show that for $$m=O\big (k+\frac{\ln {p}}{\ln {D}}\big )$$ m = O ( k + ln p ln D ) and a variety of different classes of random matrices $$H$$ H , which include the class of Gaussian matrices, existence is assured and the probability is very high. The estimate on $$m$$ m is tight in terms of $$k,p,D$$ k , p , D . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Randomized large distortion dimension reduction

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Publisher
Springer Journals
Copyright
Copyright © 2014 by Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-014-0275-7
Publisher site
See Article on Publisher Site

Abstract

Consider a random matrix $$H:{\mathbb {R}}^{n}\longrightarrow {\mathbb {R}}^{m}$$ H : R n ⟶ R m . Let $$D\ge 2$$ D ≥ 2 and let $$\{W_l\}_{l=1}^{p}$$ { W l } l = 1 p be a set of $$k$$ k -dimensional affine subspaces of $${\mathbb {R}}^{n}$$ R n . We ask what is the probability that for all $$1\le l\le p$$ 1 ≤ l ≤ p and $$x,y\in W_l$$ x , y ∈ W l , $$\begin{aligned} \Vert x-y\Vert _2\le \Vert Hx-Hy\Vert _2\le D\Vert x-y\Vert _2. \end{aligned}$$ ‖ x - y ‖ 2 ≤ ‖ H x - H y ‖ 2 ≤ D ‖ x - y ‖ 2 . We show that for $$m=O\big (k+\frac{\ln {p}}{\ln {D}}\big )$$ m = O ( k + ln p ln D ) and a variety of different classes of random matrices $$H$$ H , which include the class of Gaussian matrices, existence is assured and the probability is very high. The estimate on $$m$$ m is tight in terms of $$k,p,D$$ k , p , D .

Journal

PositivitySpringer Journals

Published: Jan 22, 2014

References

  • An elementary proof of a theorem of Johnson and Lindenstrauss
    Dasgupta, S; Gupta, A

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