Identifiability, the KL property in metric spaces, and subgradient curves

Adrian Lewis; Tonghua Tian

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Identifiability, the KL property in metric spaces, and subgradient curves

Adrian Lewis, Tonghua Tian

Optimization and Control · Submitted 5 May 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The first distills the notion to a simple metric property, applicable not just in Euclidean settings but to optimization over manifolds and beyond; the second reveals analogous continuous-time behavior for subgradient descent curves.

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Identifiability, and the closely related idea of partial smoothness, unify classical active set methods and more general notions of solution structure. Diverse optimization algorithms generate iterates in discrete time that are eventually confined to identifiable sets. We present two fresh perspectives on identifiability. The first distills the notion to a simple metric property, applicable not just in Euclidean settings but to optimization over manifolds and beyond; the second reveals analogous continuous-time behavior for subgradient descent curves. The Kurdya-Lojasiewicz property typically governs convergence in both discrete and continuous time: we explore its interplay with identifiability.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Identifiability, and the closely related idea of partial smoothness, unify classical active set methods and more general notions of solution structure.	background 1.00	0.07	0.08	asserted 1.00
s1	Diverse optimization algorithms generate iterates in discrete time that are eventually confined to identifiable sets.	background 0.98	0.10	0.07	asserted 1.00
s2	We present two fresh perspectives on identifiability.	contribution 1.00	0.91	0.07	asserted 0.98
s3	The first distills the notion to a simple metric property, applicable not just in Euclidean settings but to optimization over manifolds and beyond; the second reveals analogous continuous-time behavior for subgradient descent curves.	contribution 1.00	0.87	0.33	asserted 1.00
s4	The Kurdya-Lojasiewicz property typically governs convergence in both discrete and continuous time: we explore its interplay with identifiability.	contribution 0.58	0.42	0.12	asserted 0.98

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Original abstract

Identifiability, and the closely related idea of partial smoothness, unify classical active set methods and more general notions of solution structure. Diverse optimization algorithms generate iterates in discrete time that are eventually confined to identifiable sets. We present two fresh perspectives on identifiability. The first distills the notion to a simple metric property, applicable not just in Euclidean settings but to optimization over manifolds and beyond; the second reveals analogous continuous-time behavior for subgradient descent curves. The Kurdya-Lojasiewicz property typically governs convergence in both discrete and continuous time: we explore its interplay with identifiability.

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We deduce $x(t_k) ∈ℳ$ for all large $k$, and hence, by [ref] , $∇_ℳ f(y) ∈∂f(y)$ for all points $y ∈ℳ$ near $x̅$.

Optimality, identifiability, and sensitivity · Primal lower nice functions · 0.96

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