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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 324 (25 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 40 (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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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 19 (7 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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Solution Counting Algorithms for ConstraintCentered Search Heuristics ⋆
"... Abstract. 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 tow ..."
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Cited by 14 (2 self)
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Abstract. 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. 1
Clifford algebras and approximating the permanent
 ACM STOC
, 2002
"... ABSTRACT We study approximation algorithms for the permanent of an n \Theta n (0; 1) matrix A based on the following simple idea: obtain a random matrix B by replacing each 1entry of A independently by \Sigma e, where e is a random basis element of a suitable algebra; then output j det(B)j 2. This ..."
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Cited by 14 (1 self)
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ABSTRACT We study approximation algorithms for the permanent of an n \Theta n (0; 1) matrix A based on the following simple idea: obtain a random matrix B by replacing each 1entry of A independently by \Sigma e, where e is a random basis element of a suitable algebra; then output j det(B)j 2. This estimator is always unbiased, but it may have exponentially large variance. In our first main result we show that, if we take the algebra to be a Clifford algebra of dimension polynomial in n, then we get an estimator with small variance. Hence only a constant number of trials suffices to estimate the permanent to good accuracy. The idea of using Clifford algebras is a natural extension of earlier work by Godsil and Gutman, Karmarkar et al., and Barvinok, who used the real numbers, complex numbers and quaternions respectively. The above result implies that, in principle, this approach gives a fullypolynomial randomized approximation scheme for the permanent, provided j det(B)j
Concentration of permanent estimators for certain large matrices, Annals of Applied Probability
 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
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Cited by 13 (4 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 10 (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.
New Permanent Estimators Via NonCommutative Determinants
 Lecture Notes in Pure and Applied Mathematics
, 2000
"... . We introduce a new notion of the determinant, called symmetrized determinant, for a square matrix with the entries in an associative algebra A. The monomial expansion of the symmetrized determinant is obtained from the standard expansion of the commutative determinant by averaging the products of ..."
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Cited by 9 (0 self)
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. We introduce a new notion of the determinant, called symmetrized determinant, for a square matrix with the entries in an associative algebra A. The monomial expansion of the symmetrized determinant is obtained from the standard expansion of the commutative determinant by averaging the products of entries of the matrix in all possible orders. We show that for any fixed finitedimensional associative algebra A, the symmetrized determinant of an n \Theta n matrix with the entries in A can be computed in polynomial in n time (the degree of the polynomial is linear in the dimension of A). Then, for every associative algebra A endowed with a scalar product and unbiased probability measure, we construct a randomized polynomial time algorithm to estimate the permanent of nonnegative matrices. We conjecture that if A = Mat(d; R) is the algebra of d \Theta d real matrices endowed with the standard scalar product and Gaussian measure, the algorithm approximates the permanent of a nonnegative ...