Activities This Week

In my talk, I will describe how simultaneous stable reduction and tropical geometry can be used to construct degenerations of plane curves. This is the main ingredient in a new proof for irreducibility of Severi varieties of the projective plane. The crucial feature of this construction is that it works in positive characteristic, where the other known methods fail. The talk will be a follow up on last week’s talk and is based on joint work with Xiang He and Ilya Tyomkin.

The mixing time is a fundamental quantity measuring the rate of convergence of a Markov chain towards its stationary distribution. We will discuss the problem of estimating the mixing time from one single long trajectory of observations. The reversible setting was addressed using spectral methods by Hsu et al. (2015), who left the general case as an open problem. In the reversible setting, the analysis is greatly facilitated by the fact that the Markov operator is self-adjoint, and Weyl’s inequality allows for dimension-free perturbation analysis of the empirical eigenvalues. In the absence of reversibility, the existing perturbation analysis has a worst-case exponential dependence on the number of states. Furthermore, even if an eigenvalue perturbation analysis with better dependence on the number of states were available, in the non-reversible case the connection between the spectral gap and the mixing time is not nearly as straightforward as in the reversible case. We design a procedure, using spectral methods, that allows us to overcome the loss of self-adjointness and to recover a sample size with a polynomial dependence in some natural complexity parameters of the chain. Additionally, we will present an alternative estimation procedure that moves away from spectral methods entirely and is instead based on a generalized version of Dobrushin’s contraction. Joint work with Aryeh Kontorovich.

Estimating the Mixing Time of Ergodic Markov Chains Geoffrey Wolfer, Aryeh Kontorovich - COLT2019 http://proceedings.mlr.press/v99/wolfer19a.html
https://arxiv.org/abs/1902.01224

Mixing Time Estimation in Ergodic Markov Chains from a Single Trajectory with Contraction Methods Geoffrey Wolfer - ALT2020 https://arxiv.org/abs/1912.06845

The theory of non-commutative (nc) rational functions which are regular at 0 is well known and studied, in terms of their minimal realizations: any such function admits a unique minimal realization centred at 0 and the domain of the function coincides with the invertibility set of the (resolvent of the) realization. In addition, a nc power series around the origin will be the power series expansion of a nc rational function if and only if a given Hankel matrix built from the coefficients of the given power series has a finite rank (Fliess-Kronecker).

In this talk, we present generalizations of these ideas to the case where the centre is non-scalar. In particular, we prove the existence and uniqueness of a minimal Fornasini-Marchesini realization for every nc rational function, centred at an arbitrary matrix point in its domain of regularity, and show that using this realization, one can evaluate the function on all of its domain (of matrices of all sizes).

Unlike the case of a scalar centre, the coefficients of the realization can not be chosen arbitrarily. We present necessary and sufficient conditions (called the linearized lost-abbey conditions) on the coefficients of a minimal realization centred at a matrix point, such that there exists a nc rational function which admits the realization.

This is a joint work with Victor Vinnikov.