AN INTRODUCTION TO O-MINIMAL GEOMETRY

By M. Coste. 2002

461 citations recorded by the source corpus.

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Citations in the corpus

Gradient Descent Maximizes the Margin of Homogeneous Neural Networks (2019, background): “, semi-algebraic functions, semianalytic functions, and definable functions in an o-minimal structure (Coste, 2002; van den Dries & Miller, 1996).”
Non-Archimedean Tame Topology and Stably Dominated Types (2010, background): “See [8], [33].”
Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods (2013, background): “References on functions definable in an o-minimal structure are [25, 29].”
Characterizations of Lojasiewicz inequalities: Subgradient flows, talweg, convexity (2009, background): “In finite-dimensional spaces it has been shown in [37] that (1) is satisfied by a much larger class of functions, namely, by those that are definable in an o-minimal structure [17], or even more generally by functions belonging to analytic-geometric categories [25].”
On the convergence of the proximal algorithm for nonsmooth functions involving analytic features (2008, background): “A good introduction to o-minimal structures is Coste [ 11 ].”
Characterizations of Lojasiewicz inequalities and applications (2008, methodology): “s to the case ϕ(t) = t1−θ. In ﬁnite-dimensional spaces it has been shown in [29] that (1) is satisﬁed by a much larger class of functions, namely, by those that are deﬁnable in an o-minimal structure [15], or even more generally by functions belonging to analytic-geometric categories [21]. In the me”
The [barred L]ojasiewicz Inequality for Nonsmooth Subanalytic Functions with Applications to Subgradient Dynamical Systems (2006, background): “[3], [8]) provides an important subclass of globally subanalytic sets.”
Clarke Subgradients of Stratifiable Functions (2006, background): “Let us recall briefly a few definitions concerning o-minimal structures (see for instance, Coste [7], van der Dries-Miller [11], Ta Lê Loi [33], and references therein).”
From error bounds to the complexity of first-order descent methods for convex functions (2015, background): “he following result, which is close to an observation already made in [18]. For an insight into the notion of deﬁnability of functions, a prominent example being semi-algebraicity, one is referred to [30]. Recall that coercivity of a proper lower-semicontinuous convex function deﬁned on a ﬁnite dime”
On damped second-order gradient systems (2014, background): “First recall that the derivative of a differentiable definable function is definable in the same structure, see [21].”
A nonsmooth Morse–Sard theorem for subanalytic functions (2006, background): “For the more general framework of o-minimal structures, see Dries and Miller [7] or Coste [6].”