A Bayesian level set method for geometric inverse problems

Iglesias, Marco A.; Lu, Yulong; Stuart, Andrew M.

doi:10.4171/ifb/362

Cited by 65 publications

(113 citation statements)

References 44 publications

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“…This follows using the argument introduced in a related context in [28]: assuming that a non-zero minimizer does exist leads to a contradiction upon multiplication of that minimizer by any number less than one; and zero does not achieve the infimum.…”

Section: Bayesian Level Set We Now Definementioning

confidence: 99%

Large data and zero noise limits of graph-based semi-supervised learning algorithms

Dunlop

Slepčev

Stuart

et al. 2020

Applied and Computational Harmonic Analysis

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Scalings in which the graph Laplacian approaches a differential operator in the large graph limit are used to develop understanding of a number of algorithms for semi-supervised learning; in particular the extension, to this graph setting, of the probit algorithm, level set and kriging methods, are studied. Both optimization and Bayesian approaches are considered, based around a regularizing quadratic form found from an affine transformation of the Laplacian, raised to a, possibly fractional, exponent. Conditions on the parameters defining this quadratic form are identified under which well-defined limiting continuum analogues of the optimization and Bayesian semi-supervised learning problems may be found, thereby shedding light on the design of algorithms in the large graph setting. The large graph limits of the optimization formulations are tackled through Γ−convergence, using the recently introduced T L p metric. The small labelling noise limits of the Bayesian formulations are also identified, and contrasted with pre-existing harmonic function approaches to the problem.

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Section: Bayesian Level Set We Now Definementioning

confidence: 99%

Large data and zero noise limits of graph-based semi-supervised learning algorithms

Dunlop

Slepčev

Stuart

et al. 2020

Applied and Computational Harmonic Analysis

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“…The infimum for this functional is not achieved [34], but the ensemble based methods we employ to solve the problem implicitly apply a further regularization which circumvents this issue. Example 2.3.…”

Section: Inverse Problemsmentioning

confidence: 99%

Ensemble Kalman inversion: a derivative-free technique for machine learning tasks

2019

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The standard probabilistic perspective on machine learning gives rise to empirical risk-minimization tasks that are frequently solved by stochastic gradient descent (SGD) and variants thereof. We present a formulation of these tasks as classical inverse or filtering problems and, furthermore, we propose an efficient, gradient-free algorithm for finding a solution to these problems using ensemble Kalman inversion (EKI). Applications of our approach include offline and online supervised learning with deep neural networks, as well as graph-based semi-supervised learning. The essence of the EKI procedure is an ensemble based approximate gradient descent in which derivatives are replaced by differences from within the ensemble. We suggest several modifications to the basic method, derived from empirically successful heuristics developed in the context of SGD. Numerical results demonstrate wide applicability and robustness of the proposed algorithm.

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“…(Other construction methods of G 2 have been considered. For example, see [13].) Discretize [0, T ] into m T intervals,…”

Section: Model Problemmentioning

confidence: 99%

Stabilities of shape identification inverse problems in a Bayesian framework

Kawakami

2020

Journal of Mathematical Analysis and Applications

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A general shape identification inverse problem is studied in a Bayesian framework. This problem requires the determination of the unknown shape of a domain in the Euclidean space from finite-dimensional observation data with some Gaussian random noise. Then, the stability of posterior is studied for observation data. For each point of the space, the conditional probability that the point is included in the unknown domain given the observation data is considered. The stability is also studied for this probability distribution.As a model problem for our inverse problem, a heat inverse problem is considered. This problem requires the determination of the unknown shape of cavities in a heat conductor from temperature data of some portion of the surface of the heat conductor. To apply the above stability results to this model problem, one needs the measurability and some boundedness of the forward operator. These properties are shown.

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A Bayesian level set method for geometric inverse problems

Cited by 65 publications

References 44 publications

Large data and zero noise limits of graph-based semi-supervised learning algorithms

Large data and zero noise limits of graph-based semi-supervised learning algorithms

Ensemble Kalman inversion: a derivative-free technique for machine learning tasks

Stabilities of shape identification inverse problems in a Bayesian framework

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