Seminars

LASSO, SLOPE, OSCAR, Fused LASSO, Clustered LASSO, generalized LASSO… are popular penalized estimators for which the penalty term is a polyhedral gauge. This presentation focuses on the model pattern recovery of β; namely recovering the subdifferential of a polyhedral gauge at β, where β is an unknown parameter of regression coefficients. For LASSO, when the penalty term is the ℓ1 norm, the model pattern of β only depends on the sign of β and sign recovery via LASSO estimator is actually a well known topic in the literature. Furthermore, this presentation shows that the notion of model pattern recovery is relevant for many examples of polyhedral gauge penalty. Specifically, we introduce the “path condition”: a necessary condition for model pattern recovery, via a penalized least squares estimator, with a probability larger than 1/2. One may relax this later condition using “thresholded” penalized least squares estimators; a new class of estimators generalizing thresholded LASSO. Indeed, we show that the “accessibility condition”, a condition weaker than the “path condition”, is asymptotically sufficient for model pattern recovery. It is well known that penalized estimators can be not uniquely defined and, actually, the theory of model pattern recovery is closely related to the important issue of uniqueness. In this presentation we also introduce a necessary and sufficient condition for the uniform uniqueness of penalized least squares estimators.

The general motivation standing behind this research is to understand relationships between dynamical and model-theoretic properties of definable [topological] groups and between dynamical properties of groups of automorphisms of first order structures and model-theoretic properties of the underlying theories. More specifically, our goal is to understand model-theoretic consequences of various notions of amenability.

Among the notions of amenability that we are interested in are: definable amenability of a definable group, classical amenability of a topological group, and, more generally, [weak] definable topological amenability of a definable topological group. We also introduce and study amenable theories.

The consequences of amenability that we obtain are the appropriate versions of G-compactness: for first order theories this is the equality of Lascar strong types and Kim-Pillay strong types; for definable [topological] groups this is the equality of suitably defined connected components $G^{000}$ and $G^{00}$ of the group $G$ in question.

Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and also some results about measures and measure-like functions.

My series of talks will be based on my preprint “Amenability and definability” joint with Ehud Hrushovski and Anand Pillay. In the first series of talks, I will focus on the context of definable [topological] groups; the second series will be devoted to our new notion of amenable theory.

There are deep connections between model theory of the infinitary logic and descriptive set theory: Scott analysis, the López-Escobar theorem or the Suzuki theorem are well known examples of this phenomenon. In this talk, I will present results of a research devoted to generalizing these connections to the setting of continuous infinitary logic and Polish metric structures. In particular, I will discuss a continuous counterpart of a theorem of Hjorth and Kechris characterizing essential countability of the isomorphism relation on a given Borel class of countable structures. As an application, I will give a short model-theoretic proof of a result of Kechris saying that orbit equivalence relations induced by continuous actions of locally compact Polish groups are essentially countable. This is joint work with Andreas Hallbäck and Todor Tsankov.