A nonsmooth Morse–Sard theorem for subanalytic functions

By Jérôme Bolte, Aris Daniilidis, A. S. Lewis. Journal of Mathematical Analysis and Applications, 2006

56 citations recorded by the source corpus.

Abstract According to the Morse–Sard theorem, any sufficiently smooth function on a Euclidean space remains constant along any arc of critical points. We prove here a theorem of Morse–Sard type suitable as a tool in variational analysis: we broaden the definition of a critical point to the standard notion in nonsmooth optimization, while we restrict the functions under consideration to be semialgebraic or subanalytic. We make no assumption of subdifferential regularity. Łojasiewicz-type inequalities for nonsmooth functions follow quickly from tools of the kind we develop, leading to convergence theory for subgradient dynamical systems.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/63b4c462cc3aaeecf2191ee2393388ce5e192dca

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Citations in the corpus

Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Lojasiewicz Inequality (2008, background): “More recently Bolte, Daniilidis, Lewis and Shiota have extended the Kurdyka-Lojasiewicz inequality to nonsmooth functions in the subanalytic and o-minimal setting [16, 17, 18]; see also [15].”
Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss (2020, background): “Functions deﬁned by piecewise polynomials are subanalytic and thus both ReLU and S-ReLU networks satisfy it, see Bolte et al. (2006) for a deﬁnition.”