Computing the Homology of Real Projective Sets

Cucker, Felipe; Krick, Teresa; Shub, Michael

doi:10.1007/s10208-017-9358-8

Cited by 22 publications

(55 citation statements)

References 54 publications

(83 reference statements)

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“…We observe that in contrast with the complexity analyses (of condition numbers closely related to κ aff ) in the literature (see, e.g., [2,3,[8][9][10][11]), the bounds in eorem 6.7 depend on E x ∈[−a, a] n (κ aff (f , x) n ) and not on max x ∈[−a, a] κ aff (f , x) n . Whereas the former has finite expectation (over f ), the la er has not.…”

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confidence: 72%

Plantinga-Vegter Algorithm takes Average Polynomial Time

Cucker

Ergür

Tonelli-Cueto

2019

Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation

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We exhibit a condition-based analysis of the adaptive subdivision algorithm due to Plantinga and Vegter. e first complexity analysis of the PV Algorithm is due to Burr, Gao and Tsigaridas who proved a O 2 τ d 4 log d worst-case cost bound for degree d plane curves with maximum coefficient bit-size τ .is exponential bound, it was observed, is in stark contrast with the good performance of the algorithm in practice. More in line with this performance, we show that, with respect to a broad family of measures, the expected time complexity of the PV Algorithm is bounded by O(d 7 ) for real, degree d, plane curves. We also exhibit a smoothed analysis of the PV Algorithm that yields similar complexity estimates. To obtain these results we combine robust probabilistic techniques coming from geometric functional analysis with condition numbers and the continuous amortization paradigm introduced by Burr, Krahmer and Yap. We hope this will motivate a fruitful exchange of ideas between the different approaches to numerical computation. CCS CONCEPTS•Mathematics of computing → Computations on polynomials; Interval arithmetic; • eory of computation → Design and analysis of algorithms; Computational geometry; KEYWORDS computational algebraic geometry, numerical methods, adaptive subdivision methods, isotopy of curves, complexity ACKNOWLEDGMENTSWe thank Michael Burr for useful discussions.

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confidence: 72%

Plantinga-Vegter Algorithm takes Average Polynomial Time

Cucker

Ergür

Tonelli-Cueto

2019

Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation

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“…Proof of Theorem 1.1. The computation of H * from C is described with details in [5,Proposition 4.3].…”

Section: The Algorithmmentioning

confidence: 99%

“…(i) As in [2], we direct the reader to Section 7 of [5] for an explanation, along with a proof, of the numerical stability mentioned in the statement above.…”

Section: Introductionmentioning

confidence: 99%

Computing the Homology of Semialgebraic Sets. I: Lax Formulas

2019

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We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of semialgebraic sets given by Boolean formulas. The algorithm works in weak exponential time. This means that outside a subset of data having exponentially small measure, the cost of the algorithm is single exponential in the size of the data. This extends the work in [2] to arbitrary semialgebraic sets.All previous algorithms proposed for this problem have doubly exponential complexity (and this is so for almost all input data).

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“…The spirit and the statement of our main result is very close to this previous work but the methods are substantially renewed. There is a significant overlap with [28] where we felt that the theory could be simplified ( §3.3 and §4.1.2), but the specificity of the semialgebraic case called for the application of different tools, such as the reach ( §2), continuous Newton method ( §3.2), or the relaxation of semialgebraic inequalities ( §4.2).…”

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confidence: 99%

Computing the Homology of Basic Semialgebraic Sets in Weak Exponential Time

Bürgisser

Cucker

Lairez

2018

J. ACM

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We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets which works in weak exponential time. That is, out of a set of exponentially small measure in the space of data, the cost of the algorithm is exponential in the size of the data. All algorithms previously proposed for this problem have a complexity which is doubly exponential (and this is so for almost all data). INTRODUCTIONSemialgebraic sets (that is, subsets of Euclidean spaces defined by polynomial equations and inequalities with real coefficients) come in a wide variety of shapes and this raises the problem of describing a given specimen, from the most primitive features, such as emptiness, dimension, or number of connected components, to finer ones, such as roadmaps, Euler-Poincaré characteristic, Betti numbers, or torsion coefficients.The Cylindrical Algebraic Decomposition (CAD) introduced by Collins [23] and Wüthrich [63] in the 1970's provided algorithms to compute these features that worked within time (sD) 2 O(n) where s is the number of defining equations, D a bound on their degree and n the dimension of the ambient space. Subsequently, a substantial effort was devoted to design algorithms for these problems with single exponential algebraic complexity bounds [4, and references therein], that is, bounds of the form (sD) n O(1) . Such algorithms have been found for deciding emptiness [6,37,49], for counting connected components [7,19,20,38,39], computing the dimension [9,41], the Euler-Poincaré characteristic [2], the first few Betti numbers [3], the top few Betti numbers [5] and roadmaps (embedded curves with certain topological properties) [11,21,50].As of today, however, no single exponential algorithm is known for the computation of the whole sequence of the homology groups (Betti numbers and torsion coefficients). For complex smooth projective varieties, Scheiblechner [52] has been able to provide an algorithm computing the Betti numbers (but not the torsion coefficients) in single exponential time relying on the algebraic De Rham cohomology. The same author provided a lower bound for this problem (assuming integer coefficients) in [51], where the problem is shown to be PS -hard.

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Computing the Homology of Real Projective Sets

Cited by 22 publications

References 54 publications

Plantinga-Vegter Algorithm takes Average Polynomial Time

Plantinga-Vegter Algorithm takes Average Polynomial Time

Computing the Homology of Semialgebraic Sets. I: Lax Formulas

Computing the Homology of Basic Semialgebraic Sets in Weak Exponential Time

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