Fréchet derivative

Summary

In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative used widely in the calculus of variations. Generally, it extends the idea of the derivative from real-valued functions of one real variable to functions on normed spaces. The Fréchet derivative should be contrasted to the more general Gateaux derivative which is a generalization of the classical directional derivative. The Fréchet derivative has applications to nonlinear problems throughout mathematical analysis and physical sciences, particularly to the calculus of variations and much of nonlinear analysis and nonlinear functional analysis. Definition Let V and W be normed vector spaces, and U\subseteq V be an open s

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https://en.wikipedia.org/wiki/Fr%C3%A9chet_derivative

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Fréchet means in Wasserstein space

Yoav Zemel

This work studies the problem of statistical inference for Fréchet means in the Wasserstein space of measures on Euclidean spaces,

\mathcal W_2 ( \mathbb R^d )

. This question arises naturally from the problem of separating amplitude and phase variation in point processes, analogous to a well-known problem in functional data analysis. We formulate the point process version of the problem, show that it is canonically equivalent to that of estimating Fréchet means in

\mathcal W_2 ( \mathbb R d )

, and carry out estimation by means of

M

-estimation. This approach allows to achieve consistency in a genuinely nonparametric framework, even in a sparse sampling regime. For Cox processes on the real line, consistency is supplemented by convergence rates and, in the dense sampling regime,

\sqrt n

-consistency and a central limit theorem. Computation of the Fréchet mean is challenging when the processes are multivariate, in which case our Fréchet mean estimator is only defined implicitly as the minimiser of an optimisation problem. To overcome this difficulty, we propose a steepest descent algorithm that approximates the minimiser, and show that it converges to a local minimum. Our techniques are specific to the Wasserstein space, because Hessian-type arguments that are commonly used for similar convergence proofs do not apply to that space. In addition, we discuss similarities with generalised Procrustes analysis. The key advantage of the algorithm is that it requires only the solution of pairwise transportation problems. The results in the preceding paragraphs require properties of Fréchet means in

\mathcal W_2 ( \mathbb R ^d )

whose theory is developed, supplemented by some new results. We present the tangent bundle and exploit its relation to optimal maps in order to derive differentiability properties of the associated Fréchet functional, obtaining a characterisation of Karcher means. Additionally, we establish a new optimality criterion for local minima and prove a new stability result for the optimal maps that, enhanced with the established consistency of the Fréchet mean estimator, yields consistency of the optimal transportation maps.

EPFL2017

We consider bifurcation from the line of trivial solutions for a nonlinear eigenvalue problem on a bounded open subset, Omega, of R-N with N >= 3, containing 0. The leading term is a degenerate elliptic operator of the form L(u) = del . A del u where A is an element of C((Omega) over bar) with A > 0 on (Omega) over bar {0} and lim(x -> 0) A(x)/vertical bar x vertical bar(2) is an element of (0, infinity). Solutions should satisfy u = 0 on partial derivative Omega and the energy associated with L should be finite: integral(Omega) A vertical bar del u vertical bar(2)dx < infinity. The nonlinear terms are of lower order, depending only on u and del u. Under our hypotheses the associated Nemytskii operators are not Frechet differentiable at the trivial solution u = 0. (C) 2014 Elsevier Ltd. All rights reserved.

Pergamon-Elsevier Science Ltd2015

We consider a critical point u(0) of a functional f is an element of C-1 (H, R), where H is a real Hilbert space, and formulate criteria ensuring that u(0) lies in a potential well of f without supposing that f' is Frechet differentiable at u(0). The derivative is required to be Gateaux differentiable at u(0), but positive definiteness of f ''(u(0)) does not even ensure that f has a local minimum at u(0) when f' is not Frechet differentiable at u(0). This issue is also discussed in the context of the energy functional for a parameter dependent nonlinear eigenvalue problem and then for a particular case involving a degenerate elliptic Dirichlet problem on a bounded domain in R-N. (C) 2017 Elsevier Ltd. All rights reserved.

Pergamon-Elsevier Science Ltd2017

Fréchet means in Wasserstein space

Yoav Zemel

This work studies the problem of statistical inference for Fréchet means in the Wasserstein space of measures on Euclidean spaces,

\mathcal W_2 ( \mathbb R^d )

\mathcal W_2 ( \mathbb R d )

, and carry out estimation by means of

M

\sqrt n

\mathcal W_2 ( \mathbb R ^d )

EPFL2017