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Poster Communications Year : 2023

Cloud of points as discrete measures for Gaussian models and stochastic optimization

Abstract

We consider the task of estimating functions from a restricted number of observations where the inputs are in the form of varying-size sets of vectors. A classical method in this expensive functions context is to approximate the true expensive function with a Gaussian process that relies on semi-definite positive kernels. Varying-size sets of vectors have some peculiar characteristics : the number of vectors is not fixed and the vectors are not ordered. These properties prevent the use of classical, fixed dimension, kernels. Sets of vectors can be represented as probability distributions or as vectors composed of wellchosen characteristic features. The distribution associated with each set of vectors can be either a (non-informative) discrete uniform distribution supported by the points of the set, or an empirical Gaussian distribution characterized by a mean and a covariance matrix estimated from the vectors of the set as in [5]. With this representation, we can define semi-definite positive kernels between distributions or vectors. The kernels can be classified into explicit ones, with known feature map and scalar product, and implicit kernels where the feature maps and the scalar product come mixed together. For implicit kernels, we restrict ourselves to distance substitution kernels with an exponential form. It is known that this approach for constructing kernels requires the substitute to be conditionally semi-definite negative, see [1]. In this context, it is necessary and sufficient to have an Hilbertian metric (isometric to a L 2 norm) [4]. The distances between distributions verifying the latter condition which we consider are the sliced Wasserstein distance [3], the maximum mean discrepancy, the approximated Gaussian Wasserstein (see [2]), and the Euclidean distance for vectors of features. For explicit kernels, we investigate the probability product kernel between Gaussian distributions [5] and the scalar product of embeddings of discrete uniform laws. The prediction performance of theses kernels are compared on three types of analytical functions. The first ones mimic wind-farms productions for different winds, the second function is the inertia of a set of vectors and the third is the maximin function, well-known in the design of experiments. We first analyze the predictions over sets of vectors drawn uniformly over the same rectangular search space. Afterwards, we investigate the predictions over sets of vectors modified through geometrical transformations such as dilatation, rotation and translation. We show that some geometrical properties of the wind-farm functions can be better learned by MMD-based kernels than others. As an example, the MMD-based kernel prediction together with the true function is represented in Figure 1.
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Dates and versions

hal-04102275 , version 1 (06-06-2023)

Identifiers

  • HAL Id : hal-04102275 , version 1

Cite

Babacar Sow, Rodolphe Le Riche, Sanaa Zannane, Merlin Keller, Julien Pelamatti. Cloud of points as discrete measures for Gaussian models and stochastic optimization. MASCOT-NUM2023, Apr 2023, LE CROISIC, France. ⟨hal-04102275⟩
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