Stratifications distinguées comme outil en géométrie semi-analytique

Kurdyka, Krzysztof; Łojasiewicz, Stanisław; Zurro, María-Angeles

doi:10.1007/bf02567979

Cited by 6 publications

(8 citation statements)

References 2 publications

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“…Since dim Γ = 1, by a result of Lojasiewicz's [14] (see also [13]), the set Γ is actually semianalytic. Let Γ be a subanalytic set associated to f by Lemma 2.5.…”

Section: Arc-meromorphic Mappingsmentioning

confidence: 99%

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On the non-analyticity locus of an arc-analytic function

Kurdyka¹,

Parusiński²

2011

J. Algebraic Geom.

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Let X be a real analytic manifold. A function f : X → R is called arcanalytic if it is real analytic on each real analytic arc. In real analytic geometry there are many examples of arc-analytic functions that are not real analytic. They appear while studying the arc-symmetric sets and the blow-analytic equivalence.In this paper we show that the non-analyticity locus of an arcanalytic function is arc-symmetric. We also discuss the behavior of the non-analyticity locus under blowings-up. By a result of Bierstone and Milman, an arc-analytic function f : X → R that satisfies a polynomial equation with real analytic coefficients, can be made analytic, over any relatively compact subset of X, by a sequence of blowings-up with smooth centers. We show that these centers can be chosen, at each stage of the resolution, inside the non-analyticity loci.

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“…Since dim Γ = 1, by a result of Lojasiewicz's [14] (see also [13]), the set Γ is actually semianalytic. Let Γ be a subanalytic set associated to f by Lemma 2.5.…”

Section: Arc-meromorphic Mappingsmentioning

confidence: 99%

“…Clearly we may assume that a ∈ Γ, otherwise f is analytic at a and the statement is trivial. Since dim Γ = 1, by a result of Lojasiewicz's [14] (see also [13]), the set Γ is actually semianalytic. Then there exists a neighborhood V of a and an analytic function ψ : V ′ → R, ψ ≡ 0, which vanishes on V ′ ∩ Γ.…”

Section: Arc-meromorphic Mappingsmentioning

confidence: 99%

On the non-analyticity locus of an arc-analytic function

Kurdyka¹,

Parusiński²

2011

J. Algebraic Geom.

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“…An important special case of the Łojasiewicz inequality is the Polyak-Łojasiewicz inequality [29], which corresponds to the Łojasiewicz inequality with θ = 1 2 . In [17,18], the Łojasiewicz inequality was generalized to the Kurdyka-Łojasiewicz inequality. Subsequently, the geometric properties has been intensively studied, along with convergence studies of optimization algorithms on Kurdyka-Łojasiewicz -type functions [29,21,17,18,19,5,1,23].…”

Section: Related Workmentioning

confidence: 99%

“…Łojasiewicz functions are real-valued functions that satisfy the Łojasiewicz inequality [21]. The Łojasiewicz inequality generalizes the Polyak-Łojasiewicz inequality [29], and is a special case of the Kurdyka-Łojasiewicz (KL) inequality [17,18]. Such functions may give rise to spiral gradient flow even if smoothness and convexity are assumed [9].…”

Section: Introductionmentioning

confidence: 99%

“…Previously, the understanding of Łojasiewicz functions have been advanced by many researchers [29,21,17,18,19,5,1,23]. In their seminal work, [1] proved the-state-of-the-art convergence rate for { x t − x ∞ } t , where {x t } t is the sequence generated by the proximal algorithm, x ∞ is the limit of {x t } t that is also a critical point of the objective f , and • denotes the Euclidean norm.…”

Section: Introductionmentioning

confidence: 99%

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Convergence Rates of Stochastic Zeroth-order Gradient Descent for Łojasiewicz Functions

Wang¹

2022

Preprint

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We prove almost sure convergence rates of Zeroth-order Gradient Descent (SZGD) algorithms for Łojasiewicz functions. The SZGD algorithm iterates aswhere f is the objective function that satisfies the Łojasiewicz inequality with Łojasiewicz exponent θ, ηt is the step size (learning rate), and ∇f (xt) is the approximate gradient estimated using zeroth-order information. We show that, for smooth Łojasiewicz functions, the sequence {xt} t∈N governed by SZGD converges to a bounded point x∞ almost surely, and x∞ is a critical point of f . If θ ∈ (0, 1 2 ], f (xt) − f (x∞), ∞ s=t xs − x∞ 2 and xt − x∞ ( • is the Euclidean norm) converge to zero linearly almost surely. If θ ∈ ( 1 2 , 1), then f (xt) − f (x∞) (and ∞ s=t xs+1 − xs 2 ) converges to zero at rate o t 1 1−2θ log t almost surely; xt − x∞ converges to zero at rate o t 1−θ 1−2θ log t almost surely. To the best of our knowledge, this paper provides the first almost sure convergence rate guarantee for stochastic zeroth order algorithms for Łojasiewicz functions.

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