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SMALL NORMS IN QUADRATIC FIELDS
"... The computation of units in algebraic number fields usually is a rather hard task. Therefore, families of number fields with an explicitly given system of independent units have been of interest to mathematicians in connection with the computation of ..."
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The computation of units in algebraic number fields usually is a rather hard task. Therefore, families of number fields with an explicitly given system of independent units have been of interest to mathematicians in connection with the computation of
Modular Arithmetic on Elements of Small Norm in Quadratic Fields
"... Abstract. We describe an algorithm which rapidly computes the coefficients of elements of small norm in quadratic fields modulo a positive integer. Our method requires that an approximation of the natural logarithm of that quadratic field element is known to sufficient accuracy. To demonstrate the e ..."
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Cited by 2 (0 self)
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Abstract. We describe an algorithm which rapidly computes the coefficients of elements of small norm in quadratic fields modulo a positive integer. Our method requires that an approximation of the natural logarithm of that quadratic field element is known to sufficient accuracy. To demonstrate
On the exact space complexity of sketching and streaming small norms
 In SODA
, 2010
"... We settle the 1pass space complexity of (1 ± ε)approximating the Lp norm, for real p with 1 ≤ p ≤ 2, of a lengthn vector updated in a lengthm stream with updates to its coordinates. We assume the updates are integers in the range [−M, M]. In particular, we show the space required is Θ(ε −2 log(mM ..."
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Cited by 33 (12 self)
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We settle the 1pass space complexity of (1 ± ε)approximating the Lp norm, for real p with 1 ≤ p ≤ 2, of a lengthn vector updated in a lengthm stream with updates to its coordinates. We assume the updates are integers in the range [−M, M]. In particular, we show the space required is Θ(ε −2 log
The geometry of graphs and some of its algorithmic applications
 COMBINATORICA
, 1995
"... In this paper we explore some implications of viewing graphs as geometric objects. This approach offers a new perspective on a number of graphtheoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that res ..."
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Cited by 524 (19 self)
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that respect the metric of the (possibly weighted) graph. Given a graph G we map its vertices to a normed space in an attempt to (i) Keep down the dimension of the host space and (ii) Guarantee a small distortion, i.e., make sure that distances between vertices in G closely match the distances between
A Singular Value Thresholding Algorithm for Matrix Completion
, 2008
"... This paper introduces a novel algorithm to approximate the matrix with minimum nuclear norm among all matrices obeying a set of convex constraints. This problem may be understood as the convex relaxation of a rank minimization problem, and arises in many important applications as in the task of reco ..."
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Cited by 555 (22 self)
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This paper introduces a novel algorithm to approximate the matrix with minimum nuclear norm among all matrices obeying a set of convex constraints. This problem may be understood as the convex relaxation of a rank minimization problem, and arises in many important applications as in the task
Wiretap Lattice Codes from Number Fields with no Small Norm Elements
"... Title Wiretap lattice codes from number fields with no smallnorm elements ..."
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Cited by 1 (0 self)
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Title Wiretap lattice codes from number fields with no smallnorm elements
The Power of Convex Relaxation: NearOptimal Matrix Completion
, 2009
"... This paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries. This is known as the matrix completion problem, and comes up in a great number of applications, including the famous Netflix Prize and other similar questions in collaborative filtering. In ..."
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Cited by 359 (7 self)
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This paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries. This is known as the matrix completion problem, and comes up in a great number of applications, including the famous Netflix Prize and other similar questions in collaborative filtering
Ultraconservative Online Algorithms for Multiclass Problems
 Journal of Machine Learning Research
, 2001
"... In this paper we study online classification algorithms for multiclass problems in the mistake bound model. The hypotheses we use maintain one prototype vector per class. Given an input instance, a multiclass hypothesis computes a similarityscore between each prototype and the input instance and th ..."
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Cited by 320 (21 self)
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similarityscores. We then discuss a specific online algorithm that seeks a set of prototypes which have a small norm. The resulting algorithm, which we term MIRA (for Margin Infused Relaxed Algorithm) is ultraconservative as well. We derive mistake bounds for all the algorithms and provide further analysis
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