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A PolynomialTime Approximation Algorithm for the Permanent of a Matrix with NonNegative Entries
 Journal of the ACM
, 2004
"... Abstract. We present a polynomialtime randomized algorithm for estimating the permanent of an arbitrary n ×n matrix with nonnegative entries. This algorithm—technically a “fullypolynomial randomized approximation scheme”—computes an approximation that is, with high probability, within arbitrarily ..."
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Cited by 316 (23 self)
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Abstract. We present a polynomialtime randomized algorithm for estimating the permanent of an arbitrary n ×n matrix with nonnegative entries. This algorithm—technically a “fullypolynomial randomized approximation scheme”—computes an approximation that is, with high probability, within arbitrarily small specified relative error of the true value of the permanent. Categories and Subject Descriptors: F.2.2 [Analysis of algorithms and problem complexity]: Nonnumerical
Unsupervised Language Acquisition: Theory and Practice
, 2001
"... In this thesis I present various algorithms for the unsupervised machine learning of aspects of natural languages using a variety of statistical models. The scientific object of the work is to examine the validity of the socalled Argument from the Poverty of the Stimulus advanced in favour of the p ..."
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Cited by 41 (0 self)
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In this thesis I present various algorithms for the unsupervised machine learning of aspects of natural languages using a variety of statistical models. The scientific object of the work is to examine the validity of the socalled Argument from the Poverty of the Stimulus advanced in favour of the proposition that humans have languagespecific innate knowledge. I start by examining an a priori argument based on Gold's theorem, that purports to prove that natural languages cannot be learned, and some formal issues related to the choice of statistical grammars rather than symbolic grammars. I present three novel algorithms for learning various parts of natural languages: first, an algorithm for the induction of syntactic categories from unlabelled text using distributional information, that can deal with ambiguous and rare words; secondly, a set of algorithms for learning morphological processes in a variety of languages, including languages such as Arabic with nonconcatenative morphology; thirdly an algorithm for the unsupervised induction of a contextfree grammar from tagged text. I carefully examine the interaction between the various components, and show how these algorithms can form the basis for a empiricist model of language acquisition. I therefore conclude that the Argument from the Poverty of the Stimulus is unsupported by the evidence.
Classical deterministic complexity of Edmonds’ problem and quantum entanglement
 In Proceedings of the thirtyfifth ACM symposium on Theory of computing
, 2003
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Solution Counting Algorithms for ConstraintCentered Search Heuristics
"... Constraints have played a central role in cp because they capture key substructures of a problem and efficiently exploit them to boost inference. This paper intends to do the same thing for search, proposing constraintcentered heuristics which guide the exploration of the search space toward areas ..."
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Cited by 19 (3 self)
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Constraints have played a central role in cp because they capture key substructures of a problem and efficiently exploit them to boost inference. This paper intends to do the same thing for search, proposing constraintcentered heuristics which guide the exploration of the search space toward areas that are likely to contain a high number of solutions. We first propose new search heuristics based on solution counting information at the level of individual constraints. We then describe efficient algorithms to evaluate the number of solutions of two important families of constraints: occurrence counting constraints, such as alldifferent, and sequencing constraints, such as regular. In both cases we take advantage of existing filtering algorithms to speed up the evaluation. Experimental results on benchmark problems show the effectiveness of our approach.
Hyperbolic polynomials approach to van der Waerden and SchrijverValiant like Conjectures: sharper bounds, simpler proofs and algorithmic applications
 Proc. STOC 2006, preprint math.CO/0510452
"... Let p(x1,..., xn) = p(X), X ∈ Rn be a homogeneous polynomial of degree n in n real variables, e = (1, 1,.., 1) ∈ Rn be a vector of all ones. Such polynomial p is called ehyperbolic if for all real vectors X ∈ Rn the univariate polynomial equation P(te − X) = 0 has all real roots λ1(X) ≥... ≥ λn ..."
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Cited by 18 (6 self)
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Let p(x1,..., xn) = p(X), X ∈ Rn be a homogeneous polynomial of degree n in n real variables, e = (1, 1,.., 1) ∈ Rn be a vector of all ones. Such polynomial p is called ehyperbolic if for all real vectors X ∈ Rn the univariate polynomial equation P(te − X) = 0 has all real roots λ1(X) ≥... ≥ λn(X). The number of nonzero roots {i: λi(X) ̸ = 0}  is called Rankp(X). A ehyperbolic polynomial p is called POShyperbolic if roots of vectors X ∈ Rn + with nonnegative coordinates are also nonnegative (the orthant Rn + belongs to the hyperbolic cone) and p(e)> 0. Below {e1,..., en} stands for the canonical orthogonal basis in Rn. The main results states that if p(x1, x2,..., xn) is a POShyperbolic (homogeneous) polynomial of degree n, Rankp(ei) = Ri and p(x1, x2,..., xn) ≥ ∏ 1≤i≤n xi; xi> 0, 1 ≤ i ≤ n, then the following inequality holds ∂n p(0,..., 0) ≥
A deterministic algorithm for approximating mixed discriminant and mixed volume, and a combinatorial corollary
, 2001
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Concentration of permanent estimators for certain large matrices
 THE ANNALS OF APPLIED PROBAB
, 2004
"... Let An = (aij) n i,j=1 be an n × n positive matrix with entries in [a,b], 0<a ≤ b. LetXn = ( √ aij xij) n i,j=1 be a random matrix, where {xij} are i.i.d. N(0, 1) random variables. We show that for large n, det(XT n Xn) concentrates sharply at the permanent of An, in the sense that n−1 log(det( ..."
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Cited by 11 (3 self)
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Let An = (aij) n i,j=1 be an n × n positive matrix with entries in [a,b], 0<a ≤ b. LetXn = ( √ aij xij) n i,j=1 be a random matrix, where {xij} are i.i.d. N(0, 1) random variables. We show that for large n, det(XT n Xn) concentrates sharply at the permanent of An, in the sense that n−1 log(det(XT n Xn) / per An) →n→ ∞ 0 in probability. 1. Introduction. For a set F ⊂ R and integers n ≥ m, denote by M(n, m, F) the set of n × m matrices with entries in F.PutM(n, F) = M(n, n, F).LetSnbe the symmetric group of permutations acting on {1,...,n}. ForA∈M(n, C), the permanent of A is defined as perA = ∑
Partially Supervised Learning of Morphology with Stochastic Transducers
"... In this paper I present an algorithm for the unsupervised learning of morphology using stochastic finite state transducers, in particular Pair Hidden Markov Models. The task is viewed as an alignment problem between two sets of words. A supervised model of morphology acquisition is converted to an u ..."
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Cited by 11 (2 self)
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In this paper I present an algorithm for the unsupervised learning of morphology using stochastic finite state transducers, in particular Pair Hidden Markov Models. The task is viewed as an alignment problem between two sets of words. A supervised model of morphology acquisition is converted to an unsupervised model by treating the alignment as a further hidden variable. The use of the ExpectationMaximisation algorithm for this task is studied, which leads to calculations involving the permanent of a matrix of probabilities.
Fast approximation of the permanent for very dense problems
 In SODA ’08
, 2008
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Immanants And Finite Point Processes
, 2000
"... Given a Hermitian, nonnegative definite kernel K and a character of the symmetric group on n letters, define the corresponding immanant function K [x 1 ; : : : ; xn ] := P oe (oe) Q n i=1 K(x i ; x oe(i) ), where the sum is over all permutations oe of f1; : : : ; ng. When is the sign char ..."
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Cited by 7 (2 self)
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Given a Hermitian, nonnegative definite kernel K and a character of the symmetric group on n letters, define the corresponding immanant function K [x 1 ; : : : ; xn ] := P oe (oe) Q n i=1 K(x i ; x oe(i) ), where the sum is over all permutations oe of f1; : : : ; ng. When is the sign character (resp. the trivial character), then K is a determinant (resp. permanent). The function K is symmetric and nonnegative, and, under suitable conditions, is also nontrivial and integrable with respect to the product measure \Omega n for a given measure . In this case, K can be normalised to be a symmetric probability density. The determinantal and permanental cases or this construction correspond to the fermion and boson point processes which have been studied extensively in the literature. The case where K gives rise to an orthogonal projection of L 2 () onto a finitedimensional subspace is studied here in detail. The determinantal instance of this special case has a ...