Some speculations on pairs-of-pants decompositions and Fukaya categories

This is partly a survey and partly a speculative article, concerning a particular question about Fukaya categories of symplectic manifolds. Namely, can we decompose a symplectic manifold into standard pieces, and then reconstruct its Fukaya category by gluing together categories depending only on the geometry of each piece, in a (loosely understood) sheaf-theoretic way? For this to work, some degree of control over pseudo-holomorphic curves is required, an issue which depends on the geometry of the decomposition under consideration. Sheaf-theoretic ideas have been successfully applied to the symplectic geometry of cotangent bundles, starting with the work of Fukaya-Oh [17], and followed by Kasturirangan-Oh [26, 39] and Nadler-Zaslow [35,36] (for a survey of the last-mentioned work and related ideas of Fukaya-Smith, see [19]). Recently, Kontsevich [28] has proposed a generalization to Stein manifolds whose Lagrangian skeleta have certain singularities. However, that is not quite the direction we wish to take here

Group-Wise Shrinkage Estimation in Penalized Model-Based Clustering

Finite Gaussian mixture models provide a powerful and widely employed probabilistic approach for clustering multivariate continuous data. However, the practical usefulness of these models is jeopardized in high-dimensional spaces, where they tend to be over-parameterized. As a consequence, different solutions have been proposed, often relying on matrix decompositions or variable selection strategies. Recently, a methodological link between Gaussian graphical models and finite mixtures has been established, paving the way for penalized model-based clustering in the presence of large precision matrices. Notwithstanding, current methodologies implicitly assume similar levels of sparsity across the classes, not accounting for different degrees of association between the variables across groups. We overcome this limitation by deriving group-wise penalty factors, which automatically enforce under or over-connectivity in the estimated graphs. The approach is entirely data-driven and does not require additional hyper-parameter specification. Analyses on synthetic and real data showcase the validity of our proposal