Kirszbraun’s Theorem via an Explicit Formula

Azagra, Daniel; Gruyer, Erwan Le; Mudarra, Carlos

doi:10.4153/s0008439520000314

Cited by 8 publications

(8 citation statements)

References 30 publications

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“…Suppose also that f : K → ℝ m is Lipschitz continuous, with a given constant C > 0. Then, there exists a function with the same Lipschitz constant, C , such that For a recent constructive proof of this theorem, see [87]. Finally, we have the following lemma:…”

Section: Methodsmentioning

confidence: 93%

“…Suppose also that 𝑓: 𝐾 → ℝ 𝑚 is Lipschitz continuous, with a given constant 𝐶 > 0. Then, there exists a function 𝑓 ̃: ℝ 𝑛 → ℝ 𝑚 with the same Lipschitz constant, 𝐶, such that 𝑓 ̃|𝐾 = 𝑓 For a recent constructive proof of this theorem, see [87]. Finally, we have the following lemma: Lemma 5.3.1: Under the uniform norm, 𝐶 ∞ functions are dense in the Banach spaces of continuous functions with compact domain.…”

Section: Proof Of Propositions 21 To 23mentioning

confidence: 97%

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Emergence of universal computations through neural manifold dynamics

Vicente

2023

Preprint

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There is growing evidence that many forms of neural computation may be implemented by low dimensional dynamics unfolding at the population scale. However, neither the connectivity structure nor the general capabilities of these embedded dynamical processes are currently understood. In this work, the two most common formalisms of firing-rate models are evaluated using tools from analysis, topology and nonlinear dynamics in order to provide plausible explanations for these problems. It is shown that low-rank structured connectivity predicts the formation of invariant and globally attracting manifolds in both formalisms, which generalizes existing theories to different neural models. Regarding the dynamics arising in these manifolds, it is proved they are topologically equivalent across the considered formalisms. It is also stated that under the low-rank hypothesis, dynamics emerging in neural models are universal. These include input-driven systems, which broadens previous findings. It is then explored how low-dimensional orbits can bear the production of continuous sets of muscular trajectories, the implementation of central pattern generators and the storage of memory states. It is also proved these dynamics can robustly simulate any Turing machine over arbitrary bounded memory strings, virtually endowing rate models with the power of universal computation. In addition, it is shown how the low-rank hypothesis predicts the parsimonious correlation structure observed in cortical activity. Finally, it is discussed how this theory could provide a useful tool from which to study neuropsychological phenomena using mathematical methods.

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Section: Methodsmentioning

confidence: 93%

Section: Proof Of Propositions 21 To 23mentioning

confidence: 97%

Emergence of universal computations through neural manifold dynamics

Vicente

2023

Preprint

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“…Another classical result with L = 1 is Kirszbraun's theorem [42,65], where both the target and domain spaces are Hilbert spaces. Also in this case the extension can be given by means of an explicit formula; see [4] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Absolute Lipschitz extendability and linear projection constants

Basso¹

2022

Studia Math.

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In this mostly expository article, we give streamlined proofs of several well-known Lipschitz extension theorems. We pay special attention to obtaining statements with explicit expressions for the Lipschitz extension constants. One of our main results is an explicit version of a very general Lipschitz extension theorem of Lang and Schlichenmaier. A special case of the theorem reads as follows. We prove that if X is a metric space and A ⊂ X satisfies the condition Nagata(n, c), then any 1-Lipschitz map f : A → Y to a Banach space Y admits a Lipschitz extension F : X → Y whose Lipschitz constant is at most 1000 • (c + 1) • log 2 (n + 2). By specifying to doubling metric spaces, this recovers an extension result of Lee and Naor. We also revisit another theorem of Lee and Naor by showing that if A ⊂ X consists of n points, then Lipschitz extensions as above exist with a Lipschitz constant of at most 600 • log n • (log log n) −1 .

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“…There are many proofs of this theorem and we refer the reader to [17,29,31] for proofs that use the Kuratowski-Zorn lemma and to [2,7,6] for a constructive approach. There exists also an explicit formula for the extension (see [3]). Let us also note a proof that uses Fenchel duality and Fitzpatrick functions (see [28,5]).…”

Section: Introductionmentioning

confidence: 99%

Continuity of extensions of Lipschitz maps

Ciosmak

2021

Isr. J. Math.

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We establish the sharp rate of continuity of extensions of R m -valued 1-Lipschitz maps from a subset A of R n to a 1-Lipschitz maps on R n . We consider several cases when there exists a 1-Lipschitz extension with preserved uniform distance to a given 1-Lipschitz map. We prove that if m > 1, then a given map is 1-Lipschitz and affine if and only if such a distance preserving extension exists for any 1-Lipschitz map defined on any subset of R n . This shows a striking difference from the case m = 1, where any 1-Lipschitz function has such a property. Another example where we prove it is possible to find an extension with the same Lipschitz constant and the same uniform distance to another Lipschitz map v is when the difference between the two maps takes values in a fixed one-dimensional subspace of R m and the set A is geodesically convex with respect to a Riemannian pseudo-metric associated with v.

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Kirszbraun’s Theorem via an Explicit Formula

Abstract: Let X, Y be two Hilbert spaces, E a subset of X and G : E → Y a Lipschitz mapping.

Cited by 8 publications

References 30 publications

Emergence of universal computations through neural manifold dynamics

Emergence of universal computations through neural manifold dynamics

Absolute Lipschitz extendability and linear projection constants

Continuity of extensions of Lipschitz maps

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