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282
On the Betti numbers of sign conditions
 Proc. Amer. Math. Soc
, 2005
"... Abstract. LetRbearealclosedfieldandletQand P be finite subsets of R[X1,...,Xk] such that the set P has s elements, the algebraic set Z defined by � Q∈Q Q =0hasdimensionk ′ and the elements ofQ and P have degree at most d. Foreach0≤i≤k ′ , we denote the sum of the ith Betti numbers over the realizat ..."
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Cited by 18 (11 self)
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Abstract. LetRbearealclosedfieldandletQand P be finite subsets of R[X1,...,Xk] such that the set P has s elements, the algebraic set Z defined by � Q∈Q Q =0hasdimensionk ′ and the elements ofQ and P have degree at most d. Foreach0≤i≤k ′ , we denote the sum of the ith Betti numbers over the realizations of all sign conditions of P on Z by bi(P, Q). We prove that k bi(P, Q) ≤
Revisiting Two Theorems of Curto and Fialkow on Moment Matrices
, 2004
"... We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence... ..."
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Cited by 17 (3 self)
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We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence...
Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets (Extended Abstract)
 J. COMPL
, 2004
"... We define counting #P classes #P ¡ and in the BlumShubSmale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over ¢ , or of systems of polynomial equalities over £ , respectively, turn ou ..."
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Cited by 17 (10 self)
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We define counting #P classes #P ¡ and in the BlumShubSmale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over ¢ , or of systems of polynomial equalities over £ , respectively, turn out to be natural complete problems in these classes. We investigate to what extent the new counting classes capture the complexity of computing basic topological invariants of semialgebraic sets (over ¢ ) and algebraic sets (over £). We prove that the problem to compute the (modified) Euler characteristic of semialgebraic sets is FP #P¤complete, and that the problem to compute the geometric degree of complex algebraic sets is FP #P¥complete. We also define new counting complexity classes GCR and GCC in the classical Turing model via taking Boolean parts of the classes above, and show that the problems to compute the Euler characteristic and the geometric degree of (semi)algebraic sets given by integer polynomials are complete in these classes. We complement the results in the Turing model by proving, for all k ¦ ∈ , the FPSPACEhardness of the problem of computing the kth Betti number of the set of real zeros of a given integer polynomial. This holds with respect to the singular homology as well as for the BorelMoore homology.
Different bounds on the different Betti numbers of semialgebraic sets
 Proceedings of the ACM Symposium on Computational Geometry
, 2001
"... A classic result in real algebraic geometry due to OleinikPetrovsky, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semialgebraic sets. This bound is tight as one can construct examples having that many connected components. However, till now no signif ..."
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Cited by 16 (7 self)
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A classic result in real algebraic geometry due to OleinikPetrovsky, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semialgebraic sets. This bound is tight as one can construct examples having that many connected components. However, till now no significantly better bounds were known on the individual higher Betti numbers. We prove...
On the complexity of real functions
, 2005
"... We establish a new connection between the two most common traditions in the theory of real computation, the BlumShubSmale model and the Computable Analysis approach. We then use the connection to develop a notion of computability and complexity of functions over the reals that can be viewed as an ..."
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Cited by 15 (5 self)
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We establish a new connection between the two most common traditions in the theory of real computation, the BlumShubSmale model and the Computable Analysis approach. We then use the connection to develop a notion of computability and complexity of functions over the reals that can be viewed as an extension of both models. We argue that this notion is very natural when one tries to determine just how “difficult ” a certain function is for a very rich class of functions. 1
Triangular Decomposition of SemiAlgebraic Systems
, 2010
"... Regular chains and triangular decompositions are fundamental and welldeveloped tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue: semialgebraic systems. We show that any such system can be dec ..."
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Cited by 13 (8 self)
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Regular chains and triangular decompositions are fundamental and welldeveloped tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue: semialgebraic systems. We show that any such system can be decomposed into finitely many regular semialgebraic systems. We propose two specifications of such a decomposition and present corresponding algorithms. Under some assumptions, one type of decomposition can be computed in singly exponential time w.r.t. the number of variables. We implement our algorithms and the experimental results illustrate their effectiveness.
Positive polynomials in scalar and matrix variables, the spectral theorem, and optimization
 , in vol. Structured Matrices and Dilations. A Volume Dedicated to the Memory of Tiberiu Constantinescu
"... We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second par ..."
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Cited by 13 (3 self)
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We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second part of the survey focuses on recently discovered connections between real algebraic geometry and optimization as well as polynomials in matrix variables and some control theory problems. These new applications have prompted a series of recent studies devoted to the structure of positivity and convexity in a free ∗algebra, the appropriate setting for analyzing inequalities on polynomials having matrix variables. We sketch some of these developments, add to them and comment on the rapidly growing literature.
Polynomial Learning of Distribution Families
"... Abstract—The question of polynomial learnability of probability distributions, particularly Gaussian mixture distributions, has recently received significant attention in theoretical computer science and machine learning. However, despite major progress, the general question of polynomial learnabili ..."
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Cited by 13 (0 self)
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Abstract—The question of polynomial learnability of probability distributions, particularly Gaussian mixture distributions, has recently received significant attention in theoretical computer science and machine learning. However, despite major progress, the general question of polynomial learnability of Gaussian mixture distributions still remained open. The current work resolves the question of polynomial learnability for Gaussian mixtures in high dimension with an arbitrary fixed number of components. Specifically, we show that parameters of a Gaussian mixture distribution with fixed number of components can be learned using a sample whose size is polynomial in dimension and all other parameters. The result on learning Gaussian mixtures relies on an analysis of distributions belonging to what we call “polynomial families” in low dimension. These families are characterized by their moments being polynomial in parameters and include almost all common probability distributions as well as their mixtures and products. Using tools from real algebraic geometry, we show that parameters of any distribution belonging to such a family can be learned in polynomial time and using a polynomial number of sample points. The result on learning polynomial families is quite general and is of independent interest. To estimate parameters of a Gaussian mixture distribution in high dimensions, we provide a deterministic algorithm for dimensionality reduction. This allows us to reduce learning a highdimensional mixture to a polynomial number of parameter estimations in low dimension. Combining this reduction with the results on polynomial families yields our result on learning arbitrary Gaussian mixtures in high dimensions. Index Terms—Gaussian mixture learning, polynomial learnability I.
Testing sign conditions on a multivariate polynomial and applications
 MATHEMATICS IN COMPUTER SCIENCE
"... Let f be a polynomial in Q[X1,..., Xn] of degree D. We focus on testing the emptiness and computing at least one point in each connected component of the semialgebraic set defined by f> 0 (or f < 0 or f = 0). To this end, the problem is reduced to computing at least one point in each conne ..."
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Cited by 13 (6 self)
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Let f be a polynomial in Q[X1,..., Xn] of degree D. We focus on testing the emptiness and computing at least one point in each connected component of the semialgebraic set defined by f> 0 (or f < 0 or f = 0). To this end, the problem is reduced to computing at least one point in each connected component of a hypersurface defined by f − e = 0 for e ∈ Q positive and small enough. We provide an algorithm allowing us to determine a positive rational number e which is small enough in this sense. This is based on the efficient computation of the set of generalized critical values of the mapping f: y ∈ C n → f(y) ∈ C which is the union of the classical set K0(f) of critical values of the mapping f and K∞(f) of asymptotic critical values of the mapping f. Then, we show how to use the computation of generalized critical values in order to obtain an efficient algorithm deciding the emptiness of a semialgebraic set defined by a single inequality or a single inequation. At last, we show how to apply our contribution to determining if a hypersurface contains real regular points. We provide complexity estimates for probabilistic versions of the latter algorithms which are within O(n 7 D 4n) arithmetic operations in Q. The paper ends with practical experiments showing the efficiency of our approach.
Reasoning about probabilistic sequential programs ∗
"... A complete and decidable Hoarestyle calculus for iterationfree probabilistic sequential programs is presented using a state logic with truthfunctional propositional (not arithmetical) connectives. 1 ..."
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Cited by 13 (11 self)
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A complete and decidable Hoarestyle calculus for iterationfree probabilistic sequential programs is presented using a state logic with truthfunctional propositional (not arithmetical) connectives. 1