Dmitry Trushin

Group Symmetric Robust Covariance Estimation

In this paper, we consider Tylers robust covariance M-estimator under group symmetry constraints. We assume that the covariance matrix is invariant to the conjugation action of a unitary matrix group, referred to as group symmetry. Examples of group symmetric structures include circulant, perHermitian, and proper quaternion matrices. We introduce a group symmetric version of Tylers estimator (STyler) and provide an iterative fixed point algorithm to compute it. The classical results claim that at least n=p+1 sample points in general position are necessary to ensure the existence and uniqueness of Tylers estimator, where p is the ambient dimension. We show that the STyler requires significantly less samples. In some groups, even two samples are enough to guarantee its existence and uniqueness. In addition, in the case of elliptical populations, we provide high probability bounds on the error of the STyler. These, too, quantify the advantage of exploiting the symmetry structure. Finally, these theoretical results are supported by numerical simulations.

Splitting fields and general differential Galois theory

An algebraic technique is presented that does not use results of model theory and makes it possible to construct a general Galois theory of arbitrary nonlinear systems of partial differential equations. The algebraic technique is based on the search for prime differential ideals of special form in tensor products of differential rings. The main results demonstrating the work of the technique obtained are the theorem on the constructedness of the differential closure and the general theorem on the Galois correspondence for normal extensions. Bibliography: 14 titles.

Galois Theory of Difference Equations with Periodic Parameters

We develop a Galois theory for systems of linear difference equations with periodic parameters, for which we also introduce linear difference algebraic groups. We apply this to constructively test if solutions of linear q-difference equations, with q ∈ ℂ* and q not a root of unity, satisfy any polynomial ζ-difference equations with ζ t = 1, t ≥ 1.

Gaussian and robust Kronecker product covariance estimation

We study the Gaussian and robust covariance estimation, assuming the true covariance matrix to be a Kronecker product of two lower dimensional square matrices. In both settings we define the estimators as solutions to the constrained maximum likelihood programs. In the robust case, we consider Tylers estimator defined as the maximum likelihood estimator of a certain distribution on a sphere. We develop tight sufficient conditions for the existence and uniqueness of the estimates and show that in the Gaussian scenario with the unknown mean, p / q + q / p + 2 samples are almost surely enough to guarantee the existence and uniqueness, where p and q are the dimensions of the Kronecker product factors. In the robust case with the known mean, the corresponding sufficient number of samples is max p / q , q / p + 1 .

Computing constraint sets for differential fields

Kronecker’s Theorem and Rabin’s Theorem are fundamental results about computable elds F and the decidability of the set of irreducible polynomials overF . We adapt these theorems to the setting of dierential elds K, with constrained pairs of dierential polynomials over K assuming the role of the irreducible polynomials. We prove that two of the three basic aspects of Kronecker’s Theorem remain true here, and that the reducibility in one direction (but not the other) from Rabin’s Theorem also continues to hold.

Global sections of structure sheaves of Keigher rings

Answering a question of J.~Kovacic, we show that, for any Keigher ring, its differential spectrum coincides with the differential spectrum of the ring of global sections of the structure sheaf. In particular, we obtain the answer for Ritt algebras, that is, differential rings containing the rational numbers.

Nullstellensatz over Quasifields

We investigate the least studied class of differential rings—the class of differential rings of nonzero characteristic. We present the notion of differentially closed quasifield and develop geometrical theory of differential equations in nonzero characteristic. The notions of quasivariety and its morphisms are scrutinized. Presented machinery is a basis for reduction modulo p for differential equations.

General differential Galois theory

The general Galois theory of arbitrary nonlinear partial differential equations is presented in the paper. For each system of differential equations its splitting field and the differential Galois group are defined. The main result is the theorem on the Galois correspondence for normal extensions.