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JohnsonLindenstrauss lemma for circulant matrices
"... We prove a variant of a JohnsonLindenstrauss lemma for matrices with circulant structure. This approach allows to minimise the randomness used, is easy to implement and provides good running times. The price to be paid is the higher dimension of the target space k = O(ε −2 log 3 n) instead of the c ..."
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Cited by 21 (0 self)
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We prove a variant of a JohnsonLindenstrauss lemma for matrices with circulant structure. This approach allows to minimise the randomness used, is easy to implement and provides good running times. The price to be paid is the higher dimension of the target space k = O(ε −2 log 3 n) instead
An elementary proof of the JohnsonLindenstrauss Lemma
, 1999
"... The JohnsonLindenstrauss lemma shows that a set of n points in high dimensional Euclidean space can be mapped down into an O(log n=ffl 2 ) dimensional Euclidean space such that the distance between any two points changes by only a factor of (1 \Sigma ffl). In this note, we prove this lemma using ..."
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Cited by 147 (1 self)
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The JohnsonLindenstrauss lemma shows that a set of n points in high dimensional Euclidean space can be mapped down into an O(log n=ffl 2 ) dimensional Euclidean space such that the distance between any two points changes by only a factor of (1 \Sigma ffl). In this note, we prove this lemma using
An elementary proof of the JohnsonLindenstrauss Lemma
, 1999
"... The JohnsonLindenstrauss lemma shows that a set of n points in high dimensional Euclidean space can be mapped down into an O(log n=ffl 2 ) dimensional Euclidean space such that the distance between any two points changes by only a factor of (1 \Sigma ffl). In this note, we prove this lemma using ..."
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The JohnsonLindenstrauss lemma shows that a set of n points in high dimensional Euclidean space can be mapped down into an O(log n=ffl 2 ) dimensional Euclidean space such that the distance between any two points changes by only a factor of (1 \Sigma ffl). In this note, we prove this lemma using
Chapter 19 The JohnsonLindenstrauss Lemma
, 2010
"... Dixon was alive again. Consciousness was upon him before he could get out of the way; not for him the slow, gracious wandering from the halls of sleep, but a summary, forcible ejection. He lay sprawled, too wicked to move, spewed up like a broken spidercrab on the tarry shingle of the morning. The ..."
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Dixon was alive again. Consciousness was upon him before he could get out of the way; not for him the slow, gracious wandering from the halls of sleep, but a summary, forcible ejection. He lay sprawled, too wicked to move, spewed up like a broken spidercrab on the tarry shingle of the morning. The light did him harm, but not as much as looking at things did; he resolved, having done it once, never to move his eyeballs again. A dusty thudding in his head made the scene before him beat like a pulse. His mouth had been used as a latrine by some small creature of the night, and then as its mausoleum. During the night, too, he’d somehow been on a crosscountry run and then been expertly beaten up by secret police. He felt bad. – Lucky Jim, Kingsley Amis. In this chapter, we will prove that given a set P of n points in IR d, one can reduce the dimension of the points to k = O(ε −2 log n) such that distances are 1±ε preserved. Surprisingly, this reduction is done by randomly picking a subspace of k dimensions and projecting the points into this random subspace. One way of thinking about this result is that we are “compressing ” the input of size nd (i.e., n points with d coordinates) into size O(nε −2 log n), while (approximately) preserving distances. 19.1 The BrunnMinkowski inequality For a set A ⊆ IR d, an a point p ∈ IR d {, let A + p denote the translation of A by p. Formally, A + p = q + p ∣ q ∈ A. Definition 19.1.1 For two sets A and B in IR n, let A + B denote the Minkowski sum of
Chapter 17 The JohnsonLindenstrauss Lemma
, 2010
"... Dixon was alive again. Consciousness was upon him before he could get out of the way; not for him the slow, gracious wandering from the halls of sleep, but a summary, forcible ejection. He lay sprawled, too wicked to move, spewed up like a broken spidercrab on the tarry shingle of the morning. The ..."
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Dixon was alive again. Consciousness was upon him before he could get out of the way; not for him the slow, gracious wandering from the halls of sleep, but a summary, forcible ejection. He lay sprawled, too wicked to move, spewed up like a broken spidercrab on the tarry shingle of the morning. The light did him harm, but not as much as looking at things did; he resolved, having done it once, never to move his eyeballs again. A dusty thudding in his head made the scene before him beat like a pulse. His mouth had been used as a latrine by some small creature of the night, and then as its mausoleum. During the night, too, he’d somehow been on a crosscountry run and then been expertly beaten up by secret police. He felt bad. – Kingsley Amis, Lucky Jim. In this chapter, we will prove that given a set P of n points in IR d, one can reduce the dimension of the points to k = O(ε −2 log n) and distances are 1 ± ε reserved. Surprisingly, this reduction is done by randomly picking a subspace of k dimensions and projecting the points into this random subspace. One way of thinking about this result is that we are “compressing ” the input of size nd (i.e., n points with d coordinates) into size O(nε −2 log n), while (approximately) preserving distances. 17.1 The BrunnMinkowski inequality For a set A ⊆ IR d, an a point p ∈ IR d {, let A + p denote the translation of A by p. Formally, A + p = q + p ∣ q ∈ A. Definition 17.1.1 For two sets A and B in IR n, let A + B denote the Minkowski sum of
The JohnsonLindenstrauss lemma almost characterizes Hilbert space, but not quite
"... Let X be a normed space that satisfies the JohnsonLindenstrauss lemma (JL 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 ‖xi − x j ‖ ≤ ‖L(xi) − L(x j) ‖ ≤ O(1) ..."
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Cited by 6 (0 self)
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Let X be a normed space that satisfies the JohnsonLindenstrauss lemma (JL 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 ‖xi − x j ‖ ≤ ‖L(xi) − L(x j) ‖ ≤ O(1
Accelerating feature based registration using the JohnsonLindenstrauss lemma
 in: Medical Image Computing and ComputerAssisted Intervention (MICCAI 2009), volume 5761 of Lecture Notes in Computer Science
, 2009
"... Abstract. We introduce an efficient search strategy to substantially accelerate feature based registration. Previous feature based registration algorithms often use truncated search strategies in order to achieve small computation times. Our new accelerated search strategy is based on the realizatio ..."
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Cited by 5 (2 self)
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on the realization that the search for corresponding features can be dramatically accelerated by utilizing JohnsonLindenstrauss dimension reduction. Order of magnitude calculations for the search strategy we propose here indicate that the algorithm proposed is more than a million times faster than previously
Acceleration of Randomized Kaczmarz Method via the JohnsonLindenstrauss Lemma
, 2010
"... The Kaczmarz method is an algorithm for finding the solution to an overdetermined system of linear equations Ax = b by iteratively projecting onto the solution spaces. The randomized versionputforthbyStrohmerandVershyninyieldsprovablyexponentialconvergenceinexpectation, which for highly overdetermin ..."
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Cited by 15 (3 self)
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JohnsonLindenstrauss dimension reduction technique to keep the runtime on the same order as the original randomized version, adding only extra preprocessing time. We present a series of empirical studies which demonstrate the remarkable acceleration in convergence to the solution using this modified
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