Seminars

We study the multivariate square-root lasso, a method for fitting the multivariate response (i.e. multi-task) linear regression model with dependent errors. This estimator minimizes the nuclear norm of the residual matrix plus a convex penalty. Unlike some existing methods for multivariate response linear regression, which require explicit estimates of the error covariance matrix or its inverse, the multivariate square-root lasso criterion implicitly adapts to dependent errors and is convex. To justify the use of this estimator, we establish an error bound which illustrates that like the univariate square-root lasso, the multivariate square-root lasso is pivotal with respect to the unknown error covariance matrix. Based on our theory, we propose a simple tuning approach which requires fitting the model for only a single value of the tuning parameter, e.g., does not require cross-validation. We propose two algorithms to compute the estimator: a prox-linear alternating direction method of multipliers algorithm, and an accelerated first order algorithm which can be applied in certain cases. In both simulation studies and a genomic data application, we show that the multivariate square-root lasso can outperform more computationally intensive methods which estimate both the regression coefficient matrix and error precision matrix.

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.

We prove simplicity for the automorphism groups of order and tournament expansions of homogeneous structures like the bounded Urysohn space and the random graph. In particular, we will show that the automorphism group of the linearly ordered random graph is a simple group. The talk will be based on a preprint https://arxiv.org/pdf/1908.05249.pdf joint with Filippo Calderoni and Katrin Tent.