Optimal Linear Shrinkage Estimator for Large Dimensional Precision Matrix

In this work we construct an optimal shrinkage estimator for the precision matrix in high dimensions. We consider the general asymptotics when the number of variables $p\rightarrow\infty$ and the sample size $n\rightarrow\infty$ so that $p/n\rightarrow c\in (0, +\infty)$. The precision matrix is estimated directly, without inverting the corresponding estimator for the covariance matrix. The recent results from the random matrix theory allow us to find the asymptotic deterministic equivalents of the optimal shrinkage intensities and estimate them consistently. The resulting distribution-free estimator has almost surely the minimum Frobenius loss. Additionally, we prove that the Frobenius norms of the inverse and of the pseudo-inverse sample covariance matrices tend almost surely to deterministic quantities and estimate them consistently. At the end, a simulation is provided where the suggested estimator is compared with the estimators for the precision matrix proposed in the literature. The optimal shrinkage estimator shows significant improvement and robustness even for non-normally distributed data.Comment: 26 pages, 5 figures. This version includes the case c>1 with the generalized inverse of the sample covariance matrix. The abstract was updated accordingl

Effect of spin on electron motion in a random magnetic field

We consider properties of a two-dimensional electron system in a random magnetic field. It is assumed that the magnetic field not only influences orbital electron motion but also acts on the electron spin. For calculations, we suggest a new trick replacing the initial Hamiltonian by a Dirac Hamiltonian. This allows us to do easily a perturbation theory and derive a supermatrix sigma model, which takes a form of the conventional sigma model with the unitary symmetry. Using this sigma model we calculate several correlation functions including a spin-spin correlation function. As compared to the model without spin, we get different expressions for the single-particle lifetime and the transport time. The diffusion constant turns out to be 2 times smaller than the one for spinless particles.Comment: 7 pages, revtex, result of the spin correlation function corrected, Appendix adde

The square-kagome quantum Heisenberg antiferromagnet at high magnetic fields: The localized-magnon paradigm and beyond

We consider the spin-1/2 antiferromagnetic Heisenberg model on the two-dimensional square-kagome lattice with almost dispersionless lowest magnon band. For a general exchange coupling geometry we elaborate low-energy effective Hamiltonians which emerge at high magnetic fields. The effective model to describe the low-energy degrees of freedom of the initial frustrated quantum spin model is the (unfrustrated) square-lattice spin-1/2 $XXZ$ model in a $z$-aligned magnetic field. For the effective model we perform quantum Monte Carlo simulations to discuss the low-temperature properties of the square-kagome quantum Heisenberg antiferromagnet at high magnetic fields. We pay special attention to a magnetic-field driven Berezinskii-Kosterlitz-Thouless phase transition which occurs at low temperatures.Comment: 6 figure

Estimation and inference for dependence in multivariate data

AbstractIn this paper, a new measure of dependence is proposed. Our approach is based on transforming univariate data to the space where the marginal distributions are normally distributed and then, using the inverse transformation to obtain the distribution function in the original space. The pseudo-maximum likelihood method and the two-stage maximum likelihood approach are used to estimate the unknown parameters. It is shown that the estimated parameters are asymptotical normally distributed in both cases. Inference procedures for testing the independence are also studied

Dynamic properties of the spin-1/2 XY chain with three-site interactions

We consider a spin-1/2 XY chain in a transverse (z) field with multi-site interactions. The additional terms introduced into the Hamiltonian involve products of spin components related to three adjacent sites. A Jordan-Wigner transformation leads to a simple bilinear Fermi form for the resulting Hamiltonian and hence the spin model admits a rigorous analysis. We point out the close relationships between several variants of the model which were discussed separately in previous studies. The ground-state phases (ferromagnet and two kinds of spin liquid) of the model are reflected in the dynamic structure factors of the spin chains, which are the main focus in this study. First we consider the zz dynamic structure factor reporting for this quantity a closed-form expression and analyzing the properties of the two-fermion (particle-hole) excitation continuum which governs the dynamics of transverse spin component fluctuations and of some other local operator fluctuations. Then we examine the xx dynamic structure factor which is governed by many-fermion excitations, reporting both analytical and numerical results. We discuss some easily recognized features of the dynamic structure factors which are signatures for the presence of the three-site interactions.Comment: 28 pages, 10 fugure

The Need for Inherently Privacy-Preserving Vision in Trustworthy Autonomous Systems

Vision is a popular and effective sensor for robotics from which we can derive rich information about the environment: the geometry and semantics of the scene, as well as the age, gender, identity, activity and even emotional state of humans within that scene. This raises important questions about the reach, lifespan, and potential misuse of this information. This paper is a call to action to consider privacy in the context of robotic vision. We propose a specific form privacy preservation in which no images are captured or could be reconstructed by an attacker even with full remote access. We present a set of principles by which such systems can be designed, and through a case study in localisation demonstrate in simulation a specific implementation that delivers an important robotic capability in an inherently privacy-preserving manner. This is a first step, and we hope to inspire future works that expand the range of applications open to sighted robotic systems.Comment: 7 pages, 6 figure

Metrizability of Clifford topological semigroups

We prove that a topological Clifford semigroup $S$ is metrizable if and only if $S$ is an $M$-space and the set $E=\{e\in S:ee=e\}$ of idempotents of $S$ is a metrizable $G_\delta$-set in $S$. The same metrization criterion holds also for any countably compact Clifford topological semigroup $S$.Comment: 4 page