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Béatrice Laurent

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JMLR Journal 2023 Journal Article

MMD Aggregated Two-Sample Test

Antonin Schrab
Ilmun Kim
Mélisande Albert
Béatrice Laurent
Benjamin Guedj
Arthur Gretton

We propose two novel nonparametric two-sample kernel tests based on the Maximum Mean Discrepancy (MMD). First, for a fixed kernel, we construct an MMD test using either permutations or a wild bootstrap, two popular numerical procedures to determine the test threshold. We prove that this test controls the probability of type I error non-asymptotically. Hence, it can be used reliably even in settings with small sample sizes as it remains well-calibrated, which differs from previous MMD tests which only guarantee correct test level asymptotically. When the difference in densities lies in a Sobolev ball, we prove minimax optimality of our MMD test with a specific kernel depending on the smoothness parameter of the Sobolev ball. In practice, this parameter is unknown and, hence, the optimal MMD test with this particular kernel cannot be used. To overcome this issue, we construct an aggregated test, called MMDAgg, which is adaptive to the smoothness parameter. The test power is maximised over the collection of kernels used, without requiring held-out data for kernel selection (which results in a loss of test power), or arbitrary kernel choices such as the median heuristic. We prove that MMDAgg still controls the level non-asymptotically, and achieves the minimax rate over Sobolev balls, up to an iterated logarithmic term. Our guarantees are not restricted to a specific type of kernel, but hold for any product of one-dimensional translation invariant characteristic kernels. We provide a user-friendly parameter-free implementation of MMDAgg using an adaptive collection of bandwidths. We demonstrate that MMDAgg significantly outperforms alternative state-of-the-art MMD-based two-sample tests on synthetic data satisfying the Sobolev smoothness assumption, and that, on real-world image data, MMDAgg closely matches the power of tests leveraging the use of models such as neural networks. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2023. ( edit, beta )

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EAAI Journal 2013 Journal Article

Multilayer perceptron for the learning of spatio-temporal dynamics—application in thermal engineering

Matthias De Lozzo
Patricia Klotz
Béatrice Laurent

Thermal engineering deals with the estimation of the temperature at different spatial points and different instants for a given set of boundary and initial conditions. For this purpose, the reference model is a numerical simulation model but it is time-consuming. Consequently we build a surrogate model in order to replace it. This surrogate model is a recursive multilayer perceptron, independent of the boundary conditions and parametrized by the statistical learning of multidimensional temporal trajectories computed with the reference model. It emulates the outputs of the reference model over time from the only knowledge of initial conditions and exogenous variables. Moreover this model is able to predict these outputs in steady state, even if its formulation is time-dependent. A new methodology is proposed so as to overcome the learning problem associated to the very weak number of trajectories available for the surrogate model construction. The first step attempts to build a more robust surrogate model by considering it as the average of local models resulting from the V-folds cross-validation technique. This new kind of multilayer perceptron is much more robust and accurate, in particular when the learning dataset is very small. The second step consists in the creation of a new learning dataset which is made up of each time observation coming from each trajectory. In this way, we artificially obtain a sizeable sample allowing all the classic neural networks constructions. Furthermore, many approaches exist in order to select the best hidden neurons number but most of them are costly or require a lot of observations. We consider here a non-asymptotic approach based on the minimization of a penalized criterion providing accurate results in an economical computational way. In order to calibrate precisely the penalty term, we use the slope heuristic or the dimension jump, recently introduced in a regression framework. The validation of the method is performed on a toy function. The prediction ability of the surrogate model built with the new methodology is successfully compared to usual constructions on a simplified problem and then applied to thermal engineering.

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