Séminaire Optimisation Mathématique Modèle Aléatoire et Statistique

In this presentation, a response matrix (here, species abundances) is assumed to depend on explanatory variables (here, environmental variables) supposed many and redundant, thus demanding dimension reduction. The Supervised Component-based Generalized Linear Regression (SCGLR), a Partial Least Squares-type method, is designed to extract from the explanatory variables several components jointly supervised by the set of responses. However, this methodology still has some limitations we aim to overcome in this work. The first limitation comes from the assumption that all the responses are predicted by the same explanatory space. As a second limitation, the previous works involving SCGLR assume the responses independent conditional on the explanatory variables. Again, this is not very likely in practice, especially in situations like those in ecology, where a non-negligible part of the explanatory variables could not be measured. To overcome the first limitation, we assume that the responses are partitioned into several unknown groups. We suppose that the responses in each group are predictable from an appropriate number of specific orthogonal supervised components of the explanatory variables. The second work relaxes the conditional independence assumption. A set of few latent factors models the residual covariance matrix of the responses conditional on the components. The approaches presented in this work are tested on simulation schemes, and then applied on ecology datasets.

One of the key properties of convex problems is that every stationary point is a global optimum, and nonlinear programming algorithms that converge to local optima are thus guaranteed to find the global optimum. However, some nonconvex problems possess the same property. This observation has motivated research into generalizations of convexity. This talk proposes a new generalization which we refer to as optima-invexity: the property that only one connected set of optimal solutions exists. We state conditions for optima-invexity of unconstrained problems and discuss structures that are promising for practical use, and outline algorithmic applications of these structures.