Rate optimal estimation of quadratic functionals in inverse problems with partially unknown operator and application to testing problems

Kroll, Martin

doi:10.1051/ps/2018027

Cited by 6 publications

(5 citation statements)

References 34 publications

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“…Johannes and Schwarz [JS13] and Marteu and Sapatinas [MS17], where the training data was supposed to consist of eigenfunctions of the operator T , whence the eigenvalues are observed with noise. In that setting, the different problem of estimating a quadratic functional was considered recently by Kroll [Kro19].…”

Section: Inverse Problems With Unknown Operatormentioning

confidence: 99%

Deep learning for inverse problems with unknown operator

Álamo¹

2021

Preprint

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We consider ill-posed inverse problems where the forward operator T is unknown, and instead we have access to training data consisting of functions f i and their noisy images T f i . This is a practically relevant and challenging problem which current methods are able to solve only under strong assumptions on the training set. Here we propose a new method that requires minimal assumptions on the data, and prove reconstruction rates that depend on the number of training points and the noise level. We show that, in the regime of "many" training data, the method is minimax optimal. The proposed method employs a type of convolutional neural networks (U-nets) and empirical risk minimization in order to "fit" the unknown operator. In a nutshell, our approach is based on two ideas: the first is to relate U-nets to multiscale decompositions such as wavelets, thereby linking them to the existing theory, and the second is to use the hierarchical structure of U-nets and the low number of parameters of convolutional neural nets to prove entropy bounds that are practically useful. A significant difference with the existing works on neural networks in nonparametric statistics is that we use them to approximate operators and not functions, which we argue is mathematically more natural and technically more convenient.

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Section: Inverse Problems With Unknown Operatormentioning

confidence: 99%

Deep learning for inverse problems with unknown operator

Álamo¹

2021

Preprint

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“…Let us note that by applying Markov's inequality, it can be shown that the test 1 T k > 0 with the simplified test statistic T k := q 2 k − q k (ω 2 q /v 2 q ) 2/α and k as in (2.9), also attains the minimax radius of testing ρ a q,v q (ε q ) ∨ ρ a q,v q (ϑ q σ q ). The approach of deriving radii of testing by applying Markov's inequality has for example been used in Kroll [2019]. Since we are in Section 3 also concerned with adaptive Bonferroni aggregation, we need the sharper bound given in Proposition 2.3 for the threshold constant in terms of α.…”

Section: Indirect Testing Proceduresmentioning

confidence: 99%

“…Restricting themselves to the goodness-of-fit (ϑq = 0q) testing task in the homoscedastic setting, Marteau and Sapatinas [2017] derive upper and lower bounds for the uniform radii, featuring a logarithmic gap. Treating the signal detection task and the goodness-of-fit testing task separately, Kroll [2019] establishes matching upper and lower bounds for the minimax radii of testing uniformly over null hypotheses in Θ.…”

Section: Introductionmentioning

confidence: 99%

“…considering the sequence Yq − λqϑq instead of Yq. Obviously, this is no longer possible if λq is unknown, which motivates their separate treatment in Marteau and Sapatinas [2017] and Kroll [2019]. In contrast, the reparametrisation ( Yq, Xq) with Yq := Yq − ϑqXq of the model (1.1) allows us to deal with the signal detection problem and the goodness-of-fit problem simultaneously.…”

Section: Introductionmentioning

confidence: 99%

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Adaptive minimax testing in inverse Gaussian sequence space models

Schluttenhofer,

Johannes

2020

Preprint

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In the inverse Gaussian sequence space model with additional noisy observations of the operator, we derive nonasymptotic minimax radii of testing for ellipsoid-type alternatives simultaneously for both the signal detection problem (testing against zero) and the goodness-of-fit testing problem (testing against a prescribed sequence) without any regularity assumption on the null hypothesis. The radii are the maximum of two terms, each of which only depends on one of the noise levels. Interestingly, the term involving the noise level of the operator explicitly depends on the null hypothesis and vanishes in the signal detection case.The minimax radii are established by first showing a lower bound for arbitrary null hypotheses and noise levels. For the upper bound we consider two testing procedures, a direct test based on estimating the energy in the image space and an indirect test. Under mild assumptions, we prove that the testing radius of the indirect test achieves the lower bound, which shows the minimax optimality of the radius and the test. We highlight the assumptions under which the direct test also performs optimally. Furthermore, we apply a classical Bonferroni method for making both the indirect and the direct test adaptive with respect to the regularity of the alternative. The radii of the adaptive tests are deteriorated by an additional log-factor, which we show to be unavoidable. The results are illustrated considering Sobolev spaces and mildly or severely ill-posed inverse problems.

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“…Minimax rate optimal estimation of a quadratic functional has also been analyzed in density deconvolutions and inverse regressions in Gaussian sequence models. See, for example, Butucea [2007], Butucea and Meziani [2011] and Kroll [2019]. As far as we know, there is no published work on minimax rate-optimal adaptive estimation of a quadratic functional in NPIV models yet.…”

Section: Introductionmentioning

confidence: 99%

Simple Adaptive Estimation of Quadratic Functionals in Nonparametric IV Models

Breunig¹,

Chen²

2021

Preprint

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This paper considers adaptive estimation of quadratic functionals in the nonparametric instrumental variables (NPIV) models. Minimax estimation of a quadratic functional of a NPIV is an important problem in optimal estimation of a nonlinear functional of an ill-posed inverse regression with an unknown operator using one random sample. We first show that a leave-one-out, sieve NPIV estimator of the quadratic functional proposed by Breunig and Chen [2020] attains a convergence rate that coincides with the lower bound previously derived by Chen and Christensen [2018]. The minimax rate is achieved by the optimal choice of a key tuning parameter (sieve dimension) that depends on unknown NPIV model features. We next propose a data driven choice of the tuning parameter based on Lepski's method. The adaptive estimator attains the minimax optimal rate in the severely ill-posed case and in the regular, mildly ill-posed case, but up to a multiplicative √ log n in the irregular, mildly ill-posed case.

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Rate optimal estimation of quadratic functionals in inverse problems with partially unknown operator and application to testing problems

Cited by 6 publications

References 34 publications

Deep learning for inverse problems with unknown operator

Deep learning for inverse problems with unknown operator

Adaptive minimax testing in inverse Gaussian sequence space models

Simple Adaptive Estimation of Quadratic Functionals in Nonparametric IV Models

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