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## Sparse covariance matrix estimation in high-dimensional deconvolution

We study the estimation of the covariance matrix Σ of a p-dimensional nor- mal random vector based on n independent observations corrupted by additive noise. Only a general nonparametric assumption is imposed on the distribution of the noise without any sparsity constraint on its covariance matrix. In this high-dimensional semiparametric deconvolution problem, we propose spectral thresholding estimators that are adaptive to the sparsity of Σ. We establish an oracle inequality for these estimators under model miss- specification and derive non-asymptotic minimax convergence rates that are shown to be logarithmic in log p/n. We also discuss the estimation of low-rank matrices based on indi- rect observations as well as the generalization to elliptical distributions. The finite sample performance of the threshold estimators is illustrated in a numerical example.

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 Tyler’s 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.

We study a problem of designing an optimal two-dimensional circularly symmetric convolution kernel (or point spread function (PSF)) with a circular support of a chosen radius *R*. Such function will be optimal for estimating an unknown signal (image) from an observation obtained through a convolution-type distortion with the additive random noise. This technique is then generalized to the case of an imprecisely known or random PSF of the measurement distortion. It is shown that the construction of the optimal convolution kernel reduces to a one-dimensional Fredholm equation of the first or a second kind on the interval [0,*R*]. If the reconstruction PSF is sought in a finite-dimensional class of functions, the problem naturally reduces to a finite-dimensional optimization problem or even a system of linear equations. We also analyze how reconstruction quality depends on the radius of the convolution kernel. It allows finding a good balance between computational complexity and quality of the image reconstruction.

In this paper, we consider Tyler's 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 Tyler's 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 Tyler's 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.

Despite the increasingly broad use of perfusion applications, we still have no generally accessible means for their verification: The common sense of perfusion maps and "bona fides" of perfusion software vendors remain the only grounds for acceptance. Thus, perfusion applications are one of a very few clinical tools considerably lacking practical objective hands-on validation. MATERIALS AND METHODS. To solve this problem, we introduce digital perfusion phantoms (DPPs) - numerically simulated DICOM image sequences specifically designed to have known perfusion maps with simple visual patterns. Processing DPP perfusion sequences with any perfusion algorithm or software of choice and comparing the results with the expected DPP patterns provide a robust and straightforward way to control the quality of perfusion analysis, software, and protocols. RESULTS. The deviations from the expected DPP maps, observed in each perfusion software, provided clear visualization of processing differences and possible perfusion implementation errors. CONCLUSION. Perfusion implementation errors, often hidden behind real-data anatomy and noise, become very visible with DPPs. We strongly recommend using DPPs to verify the quality of perfusion applications.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.