Application of Methods of Ordinary Differential Equations to Global Inverse Function Theorems

Arutyunov, A. V.; Zhukovskiy, S. E.

doi:10.1134/s0012266119040013

Cited by 13 publications

(10 citation statements)

References 12 publications

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“…To this aim, we check the assumptions of Theorem 6 in Arutyunov and Zhukovskiy [2019] for the function F . The desired conclusion will follow.…”

Section: Discussionmentioning

confidence: 99%

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What's a good imputation to predict with missing values?

Morvan¹,

Josse²,

Scornet³

et al. 2021

Preprint

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How to learn a good predictor on data with missing values? Most efforts focus on first imputing as well as possible and second learning on the completed data to predict the outcome. Yet, this widespread practice has no theoretical grounding. Here we show that for almost all imputation functions, an impute-then-regress procedure with a powerful learner is Bayes optimal. This result holds for all missing-values mechanisms, in contrast with the classic statistical results that require missing-at-random settings to use imputation in probabilistic modeling. Moreover, it implies that perfect conditional imputation may not be needed for good prediction asymptotically. In fact, we show that on perfectly imputed data the best regression function will generally be discontinuous, which makes it hard to learn. Crafting instead the imputation so as to leave the regression function unchanged simply shifts the problem to learning discontinuous imputations. Rather, we suggest that it is easier to learn imputation and regression jointly. We propose such a procedure, adapting NeuMiss, a neural network capturing the conditional links across observed and unobserved variables whatever the missing-value pattern. Experiments confirm that joint imputation and regression through NeuMiss is better than various two step procedures in our experiments with finite number of samples.Preprint. Under review.

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“…To this aim, we check the assumptions of Theorem 6 in Arutyunov and Zhukovskiy [2019] for the function F . The desired conclusion will follow.…”

Section: Discussionmentioning

confidence: 99%

“…Since f is uniformly continuous and twice continuously differentiable, condition 1 − 3 of Theorem 6 in Arutyunov and Zhukovskiy [2019] are satisfied. To verify the next condition, we have to prove that Derivation of the Bayes predictor for f bowl .…”

Section: Discussionmentioning

confidence: 99%

What's a good imputation to predict with missing values?

Morvan¹,

Josse²,

Scornet³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The Banach open mapping theorem means that > 0 if and only if The Michael selection theorem (see [5]) guarantees that, if Theorem 1 is proved by applying methods of the theory of ordinary differential equations; more specifically, the proof consists in an analysis of solutions to some Cauchy problem based on the mapping f (see [3]).…”

Section: Given a Linear Operator We Definementioning

confidence: 99%

“…Let us extend the result of [3] to Hilbert spaces. For this purpose, we first introduce the necessary concepts.…”

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confidence: 99%

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On Global Solvability of Nonlinear Equations with Parameters

Arutyunov

Zhukovskiy

2021

Dokl. Math.

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We consider smooth mappings acting from one Banach space to another and depending on a parameter belonging to a topological space. Under various regularity assumptions, sufficient conditions for the existence of global and semilocal continuous inverse and implicit functions are obtained. We consider applications of these results to the problem of continuous extension of implicit functions and to the problem of coincidence points of smooth and continuous compact mappings.

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