The Johnson-Lindenstrauss lemma and the sphericity of some graphs

Frankl, Péter; Maehara, Hiroshi

doi:10.1016/0095-8956(88)90043-3

Cited by 223 publications

(153 citation statements)

References 10 publications

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“…Work has been done on finding suitable constructions of such mappings f (e.g. Frankl and Maehara, 1988;Achlioptas, 2003). In our experiments, we have employed a commonly used mapping (R) which consists of the vectors of normally distributed N(0,1) random numbers and the row vectors of the random matrix are scaled to have unit length.…”

Section: Random Projectionsmentioning

confidence: 99%

Random projections in reducing the dimensionality of climate simulation data

Seitola

Mikkola

Silén

et al. 2014

Tellus A: Dynamic Meteorology and Oceanography

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A B S T R A C T Random projection (RP) is a dimensionality reduction method that has been earlier applied to high-dimensional data sets, for instance, in image processing. This study presents experimental results of RP applied to simulated global surface temperature data. Principal component analysis (PCA) is utilised to analyse how RP preserves structures when the original data set is compressed down to 10% or 1% of its original volume. Our experiments show that, although information is naturally lost in RP, the main spatial patterns (the principal component loadings) and temporal signatures (spectra of the principal component scores) can nevertheless be recovered from the randomly projected low-dimensional subspaces. Our results imply that RP could be used as a preprocessing step before analysing the structure of high-dimensional climate data sets having many state variables, time steps and spatial locations.

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Section: Random Projectionsmentioning

confidence: 99%

Random projections in reducing the dimensionality of climate simulation data

Seitola

Mikkola

Silén

et al. 2014

Tellus A: Dynamic Meteorology and Oceanography

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show abstract

“…Here, the target dimension k is much smaller than the original dimension d. The name random projections was coined after the first construction by Johnson and Lindenstrauss in [1] who showed that such mappings exist for k ∈ O(log(1/δ)/ε 2 ). Other constructions of random projection matrices have been discovered since [2,3,4,5,6]. Their properties make random projections a key player in rank-k approximation algorithms [7,8,9,10,11,12,13,14], other algorithms in numerical linear algebra [15,16,17], compressed sensing [18,19,20], and various other applications, e.g, [21,22].…”

Section: Introductionmentioning

confidence: 99%

“…entries such as in [1,2,3,4,5,6] would guarantee the desired low distortion property for the entire space of matrix, D s . 4 In their analysis they show that Ψ = AD s is a good random projection for vectors x with bounded 4 norm.…”

Section: Introductionmentioning

confidence: 99%

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Dense Fast Random Projections and Lean Walsh Transforms

Liberty

Ailon

Singer

2010

Discrete Comput Geom

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Random projection methods give distributions over k × d matrices such that if a matrix Ψ (chosen according to the distribution) is applied to a finite set of vectors xi ∈ R d the resulting vectors Ψxi ∈ R k approximately preserve the original metric with constant probability. First, we show that any matrix (composed with a random ±1 diagonal matrix) is a good random projector for a subset of vectors in R d . Second, we describe a family of tensor product matrices which we term Lean Walsh. We show that using Lean Walsh matrices as random projections outperforms, in terms or running time, the best known current result (due to Matousek) under comparable assumptions.

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“…The JL-lemma has been shown to be useful in applications in computer science and in engineering, and various proofs were given in [1,3,7,10,16].…”

Section: Introduction In 1984 Johnson and Lindenstraussmentioning

confidence: 99%

Generalizing the Johnson–Lindenstrauss lemma to k-dimensional affine subspaces

Dmitriyuk

Gordon

2009

Studia Math.

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Abstract. Let ε > 0 and 1 ≤ k ≤ n and let {W l } p l=1 be affine subspaces of R n , each of dimension at most k. Let m = O(ε −2 (k + log p)) if ε < 1, and m = O(k + log p/log(1 + ε)) if ε ≥ 1. We prove that there is a linear map H : R n → R m such that for all 1 ≤ l ≤ p and x, y ∈ W l we have x − y 2 ≤ H(x) − H(y) 2 ≤ (1 + ε) x − y 2, i.e. the distance distortion is at most 1 + ε. The estimate on m is tight in terms of k and p whenever ε < 1, and is tight on ε, k, p whenever ε ≥ 1. We extend these results to embeddings into general normed spaces Y .

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The Johnson-Lindenstrauss lemma and the sphericity of some graphs

Cited by 223 publications

References 10 publications

Random projections in reducing the dimensionality of climate simulation data

Random projections in reducing the dimensionality of climate simulation data

Dense Fast Random Projections and Lean Walsh Transforms

Generalizing the Johnson–Lindenstrauss lemma to k-dimensional affine subspaces

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