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

Abstract

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 .

Uniform estimates for order statistics and Orlicz functions

“…[8,[10][11][12]14,17,24,[28][29][30]). This list of course does not include the enormous quantity of published works which deal with evaluations and applications of max and min associated with various random parameters, e.g., smallest and largest eigenvalues of random matrices, as these are the extreme values in the scale of order statistics.…”

“…To name only a few: Wireless networks, signal processing, image processing, compressed sensing, data reconstruction, learning theory and data mining. A sample of works done in this area are [2,4,6,8,9,10,12,13,17,31,33].…”

Metric Dimension Reduction: A Snapshot of the Ribe Program

“…Remark 37. The JL lemma was reproved many times; see [94,102,121,23,75,2,136,122,166,3,82,138,4,83,133,49,81], though we make no claim that this is a comprehensive list of references. There were several motivations for these further investigations, ranging from the desire to obtain an overall better understanding of the JL phenomenon, to obtain better bounds, and to obtain distributions on random matrices A as in (31) with certain additional properties that are favorable from the computational perspective, such as ease of simulation, use of fewer random bits, sparsity, and the ability to evaluate the mapping (z ∈ R n−1 ) → Az quickly (akin to the fast Fourier transform).…”

Large distortion dimension reduction using random variable

“…standard Gaussian entries. We recall another of the main results in [7] which holds for Gaussian matrices G for large and small distortions: Theorem 1.1 There is a positive constant c such that the following holds: Given 0 < ε < ∞ and 1 ≤ k ≤ n and p subspaces {W l } p l=1 of dimension at most k in ℓ n 2 and any m ≥ c (1 + ε −2 )k + 1+ε ε ln(1+ε) ln p and a Gaussian m × n matrix G with i.i.d. standard Gaussian entries, there is a number E such that the probability that for every 1 ≤ l ≤ p and x, y ∈ W l…”

“…The paper is concerned with large controlled distance distortion of preassigned magnitude D by random linear maps H ∈ L(R n , R m ) where H is a matrix whose rows satisfy certain conditions. To cite one example of large distortion is Theorem 2.8 in [7] where G = (g j,i ) m j=1 n i=1 is a Gaussian matrix with i.i.d. standard Gaussian entries.…”