On gradients of functions definable in o-minimal structures

By K. Kurdyka. 1998

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Citations in the corpus

Proximal alternating linearized minimization for nonconvex and nonsmooth problems (2013, unknown): “Forthereader’sconvenience,otherrelated factsandpertinentresultsarealsosummarizedinthesameappendix.”
A Globally Convergent Algorithm for Nonconvex Optimization Based on Block Coordinate Update (2014, background): “Kurdyka extended this property to functions on the o-minimal structure in [27].”
Projection-Free Optimization on Uniformly Convex Sets (2020, background): “When the function f is subanalytic, (HEB) is known to hold generically (Lojasiewicz, 1965; Kurdyka, 1998; Bolte et al., 2007).”
Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods (2013, background): “The monographs [26, 30] are good references on o-minimal structures; concerning Kurdyka- Lojasiewicz inequalities in this context the reader is referred to [38, 17].”
On damped second-order gradient systems (2014, methodology): “[43, 44]), it was generalized to o-minimal structures and considerably simplified by Kurdyka in [42] (see Appendix).”
On Convergence of Model Parallel Proximal Gradient Algorithm for Stale Synchronous Parallel System (2016, methodology): “Then, inspired by the recent KurdykaLojasiewicz (K L) inequality [3, 5, 9, 11, 20], we further prove in Theorem 2 that the whole sequence of msPG in fact converges to a critical point, under mild technical assumptions that we verify for many familiar examples.”
iPiano: Inertial Proximal Algorithm for Nonconvex Optimization (2014, background): “However, with slightly more assumptions to the objective, namely, that it satisfies the Kurdyka– Lojasiewicz inequality [30, 31, 26], several algorithms have been shown to converge [14, 5, 3, 4].”
Convergence Analysis of a Momentum Algorithm with Adaptive Step Size for Non Convex Optimization (2019, background): “2), feedforward neural networks with arbitrary number of layers of arbitrary dimensions, with activations such as sigmoid, ReLU, leaky ReLU, tanh, softplus (and many others), with a loss function such as lp norm, hinge loss, logistic loss or cross entropy (and many others), belong to this class of s”
Modern regularization methods for inverse problems (2018, background): “In a finite dimensional setting U = Rn it is now sufficient to assume that E + J is a KurdykaLojasiewicz (KL) function [270, 255, 45] in order to show that the iterates (85) converge globally to a critical point of E + J .”
Boosting Frank-Wolfe by Chasing Gradients (2020, background): “See also the Kurdyka-Łojasiewicz inequality (Kurdyka, 1998; Łojasiewicz, 1963) for a generalization to nonsmooth optimization (Bolte et al., 2017).”
Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Lojasiewicz Inequality (2008, background): “In a recent paper, Kurdyka [35] has extended this result to differentiable functions definable in an o-minimal structure (see Section 4.”
Characterizations of Lojasiewicz inequalities and applications (2008, background): “Early connections between the K Linequality and this old concept can be found in [29], and even more clearly in [16].”
Clarke Subgradients of Stratifiable Functions (2006, background): “It is worthwhile pointing out that, even in a smooth setting, this last result generalizes the one given in [19] by removing the boundedness assumption on the domain of the function.”
Distributed block-diagonal approximation methods for regularized empirical risk minimization (2017, methodology): “We demonstrate that when the objective function satisﬁes the Kurdyka-Łojasiewicz inequality (Łojasiewicz 1963, 1993; Kurdyka 1998) 5 , the proposed algorithm converge globally linearly.”