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424
Training Linear SVMs in Linear Time
, 2006
"... Linear Support Vector Machines (SVMs) have become one of the most prominent machine learning techniques for highdimensional sparse data commonly encountered in applications like text classification, wordsense disambiguation, and drug design. These applications involve a large number of examples n ..."
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Cited by 549 (6 self)
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as well as a large number of features N, while each example has only s << N nonzero features. This paper presents a CuttingPlane Algorithm for training linear SVMs that provably has training time O(sn) for classification problems and O(sn log(n)) for ordinal regression problems. The algorithm
Nondeterministic Space is Closed Under Complementation
, 1988
"... this paper we show that nondeterministic space s(n) is closed under complementation, for s(n) greater than or equal to log n. It immediately follows that the contextsensitive languages are closed under complementation, thus settling a question raised by Kuroda in 1964 [9]. See Hartmanis and Hunt [4 ..."
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Cited by 262 (14 self)
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this paper we show that nondeterministic space s(n) is closed under complementation, for s(n) greater than or equal to log n. It immediately follows that the contextsensitive languages are closed under complementation, thus settling a question raised by Kuroda in 1964 [9]. See Hartmanis and Hunt
Compressed representations of sequences and fulltext indexes
 ACM Transactions on Algorithms
, 2007
"... Abstract. Given a sequence S = s1s2... sn of integers smaller than r = O(polylog(n)), we show how S can be represented using nH0(S) + o(n) bits, so that we can know any sq, as well as answer rank and select queries on S, in constant time. H0(S) is the zeroorder empirical entropy of S and nH0(S) pro ..."
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Cited by 155 (76 self)
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Abstract. Given a sequence S = s1s2... sn of integers smaller than r = O(polylog(n)), we show how S can be represented using nH0(S) + o(n) bits, so that we can know any sq, as well as answer rank and select queries on S, in constant time. H0(S) is the zeroorder empirical entropy of S and nH0(S
THE SHORTESTPATH PROBLEM FOR GRAPHS WITH RANDOM ARCLENGTHS
, 1985
"... We consider the problem of finding the shortest distance between all pairs of vertices in a complete digraph on n vertices, whose arclengths are nonnegative random variables. We describe an algorithm which solves this problem in O(n(m + n log n)) expected time, where m is the expected number of ar ..."
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Cited by 102 (5 self)
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the town; at each stage, each person who knows the rumour tells someone else, chosen randomly from the town and independently of all other choices. Let Sn be the number of stages before the whole town knows the rumour. We show that Sn/log2n " 1 + loge 2 in probability as n ~ 0 % and estimate
Sparsistency and rates of convergence in large covariance matrices estimation
, 2009
"... This paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions. Here, sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probabi ..."
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Cited by 110 (12 self)
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for these problems under the Frobenius norm are of order (sn log pn/n) 1/2, where sn is the number of nonzero elements, pn is the size of the covariance matrix and n is the sample size. This explicitly spells out the contribution of highdimensionality is merely of a logarithmic factor. The conditions on the rate
SN(α):= N∑
, 2008
"... n=1 as N → ∞ is not transparent. The random walk ∑N n=1 wn, where the wn are independent random variables taking the values ±1 with equal probability, is known [22] to typically have absolute value around c √ N, for an appropriate constant c and large N. Knowing this, and knowing that for irrationa ..."
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that for irrational α the sequence ⌊nα ⌋ is “randomish ” modulo 2, a natural guess is that SN(α)  is also around √ N. Contrary to this expectation, for almost all real numbers α ∣SN(α) ∣ 2 ≤ (log N) (1) for all large N. This is a corollary of a theorem of Khintchine, which we state precisely in Section 1. We
Fast algorithms for componentbycomponent construction of rank1 lattice rules in shiftinvariant reproducing kernel Hilbert spaces
 Math. Comp
, 2004
"... Abstract. We reformulate the original componentbycomponent algorithm for rank1 lattices in a matrixvector notation so as to highlight its structural properties. For function spaces similar to a weighted Korobov space, we derive a technique which has construction cost O(sn log(n)),incontrastwitht ..."
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Cited by 51 (10 self)
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Abstract. We reformulate the original componentbycomponent algorithm for rank1 lattices in a matrixvector notation so as to highlight its structural properties. For function spaces similar to a weighted Korobov space, we derive a technique which has construction cost O(sn log
The largest prime dividing the maximal order of an element of Sn
 Math. Comp
, 1995
"... Abstract. We define g(n) to be the maximal order of an element of the symmetric group on n elements. Results about the prime factorization of g(n) allow a reduction of the upper bound on the largest prime divisor of g(n) to 1.328 V " log"Let S „ be the symmetric group on « letters. Defini ..."
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Cited by 5 (0 self)
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. Definition. g(n) = max {ord(tr)  a e Sn}. The first work on g(n) was done by Landau [1] in 1903. He showed that logg(«) ~ \Jn log « as « —> oo. In 1984, Massias [2] showed an upper bound for xJS(n) yjn log n ' logg(«) < ayjnlogn a = 1.05313..., «>1, with a attained for « = 1, 319, 166. Let
Logdet0 – a MATLAB software for solving logdeterminant optimization problems
"... logdeterminant optimization problems of the form: min X {〈C, X 〉 − µ log det X: A(X) = b, X ≽ 0}, where C ∈ Sn, b ∈ Rm, µ ≥ 0 is a given parameter, A: Sn → Rm is a linear map. Note that the linear map A can be expressed as ..."
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logdeterminant optimization problems of the form: min X {〈C, X 〉 − µ log det X: A(X) = b, X ≽ 0}, where C ∈ Sn, b ∈ Rm, µ ≥ 0 is a given parameter, A: Sn → Rm is a linear map. Note that the linear map A can be expressed as
On some problems of a statistical group theory V
, 1971
"... In the second paper of this series (see [2]) we dealt with statistical theorems concerning the arithmetical structure of O(P) the order of the element P in symmetric group S „ of n letters. If w(x) / with x arbitrarily slowly and assigning to the phrase "for almost all P's " the me ..."
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Cited by 76 (0 self)
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; the meaning "for all but o(n!) P's " the theorems in question run as follows. THEOREM A. For almost all P's the order 0(P) is divisible by all prime powers not exceeding (1 1) log n 1 + 3 log log log n w (n) log log n log log n log log n} The theorem is best possible in the strong sense
Results 1  10
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424