Quantized Low-Rank Multivariate Regression with Random Dithering

Low-rank multivariate regression (LRMR) is an important statistical learning model that combines highly correlated tasks as a multiresponse regression problem with low-rank priori on the coefficient matrix. In this paper, we study quantized LRMR, a practical setting where the responses and/or the covariates are discretized to finite precision. We focus on the estimation of the underlying coefficient matrix. To make consistent estimator that could achieve arbitrarily small error possible, we employ uniform quantization with random dithering, i.e., we add appropriate random noise to the data before quantization. Specifically, uniform dither and triangular dither are used for responses and covariates, respectively. Based on the quantized data, we propose the constrained Lasso and regularized Lasso estimators, and derive the non-asymptotic error bounds. With the aid of dithering, the estimators achieve minimax optimal rate, while quantization only slightly worsens the multiplicative factor in the error rate. Moreover, we extend our results to a low-rank regression model with matrix responses. We corroborate and demonstrate our theoretical results via simulations on synthetic data or image restoration.Comment: 16 pages (Submitted

Theory of conserved spin current and its application to two dimensional hole gas

We present a detailed microscopic theory of the conserved spin current which is introduced by us [Phys. Rev. Lett. \textbf{96}, 196602 (2006)] and satisfies the spin continuity equation even for spin-orbit coupled systems. The spin transport coefficients $\sigma_{\mu\nu}^{s}$ as a response to the electric field are shown to consist of two parts, i.e., the conventional part $\sigma_{\mu\nu}^{s0}$ and the spin torque dipole correction $\sigma_{\mu\nu }^{s\tau}$. As one key result, an Onsager relation between $\sigma_{\mu\nu }^{s}$ and other kinds of transport coefficients are shown. The expression for $\sigma_{\mu\nu}^{s}$ in terms of single-particle Bloch states are derived, by use of which we study the conserved spin Hall conductivity in the two dimensional hole gas modeled by a combined Luttinger and SIA Rashba spin-orbit coupling. It is shown that the two components in spin Hall conductivity usually have the opposite contributions. While in the absence of Rashba spin splitting, the spin Hall transport is dominated by the conventional contribution, the presence of Rashba spin splitting stirs up a large enhancement of the spin torque dipole correction, leading to an overall sign change for the total spin Hall conductivity. Furthermore, an approximate two-band calculation and the subsequent comparison with the exact four-band results are given, which reveals that the coupling between the heavy hole and light hole bands should be taken into account for strong Rashba spin splitting.Comment: 10 pages, 4 figure

Magnetic control of the valley degree of freedom of massive Dirac fermions with application to transition metal dichalcogenides

We study the valley-dependent magnetic and transport properties of massive Dirac fermions in multivalley systems such as the transition metal dichalcogenides. The asymmetry of the zeroth Landau level between valleys and the enhanced magnetic susceptibility can be attributed to the different orbital magnetic moment tied with each valley. This allows the valley polarization to be controlled by tuning the external magnetic field and the doping level. As a result of this magnetic field induced valley polarization, there exists an extra contribution to the ordinary Hall effect. All these effects can be captured by a low energy effective theory with a valley-orbit coupling term.Comment: 9 pages, 6 figure

High Dimensional Statistical Estimation under Uniformly Dithered One-bit Quantization

In this paper, we propose a uniformly dithered 1-bit quantization scheme for high-dimensional statistical estimation. The scheme contains truncation, dithering, and quantization as typical steps. As canonical examples, the quantization scheme is applied to the estimation problems of sparse covariance matrix estimation, sparse linear regression (i.e., compressed sensing), and matrix completion. We study both sub-Gaussian and heavy-tailed regimes, where the underlying distribution of heavy-tailed data is assumed to have bounded moments of some order. We propose new estimators based on 1-bit quantized data. In sub-Gaussian regime, our estimators achieve near minimax rates, indicating that our quantization scheme costs very little. In heavy-tailed regime, while the rates of our estimators become essentially slower, these results are either the first ones in an 1-bit quantized and heavy-tailed setting, or already improve on existing comparable results from some respect. Under the observations in our setting, the rates are almost tight in compressed sensing and matrix completion. Our 1-bit compressed sensing results feature general sensing vector that is sub-Gaussian or even heavy-tailed. We also first investigate a novel setting where both the covariate and response are quantized. In addition, our approach to 1-bit matrix completion does not rely on likelihood and represent the first method robust to pre-quantization noise with unknown distribution. Experimental results on synthetic data are presented to support our theoretical analysis.Comment: We add lower bounds for 1-bit quantization of heavy-tailed data (Theorems 11, 14

Orbital magnetization and its effect in antiferromagnets on the distorted fcc lattice

We study the intrinsic orbital magnetization (OM) in antiferromagnets on the distorted face-centered-cubic lattice. The combined lattice distortion and spin frustration induce nontrivial $k$-space Chern invariant, which turns to result in profound effects on the OM properties. We derive a specific relation between the OM and the Hall conductivity, according to which it is found that the intrinsic OM vanishes when the electron chemical potential lies in the Mott gap. The distinct behavior of the intrinsic OM in the metallic and insulating regions is shown. The Berry phase effects on the thermoelectric transport is also discussed.Comment: 18 pages, 6 figure

Magnetoelectric Coupling and Electric Control of Magnetization in Ferromagnet-Ferroelectric-Metal Superlattices

Ferromagnet-ferroelectric-metal superlattices are proposed to realize the large room-temperature magnetoelectric effect. Spin dependent electron screening is the fundamental mechanism at the microscopic level. We also predict an electric control of magnetization in this structure. The naturally broken inversion symmetry in our tri-component structure introduces a magnetoelectric coupling energy of $P M^2$. Such a magnetoelectric coupling effect is general in ferromagnet-ferroelectric heterostructures, independent of particular chemical or physical bonding, and will play an important role in the field of multiferroics.Comment: 5 pages including 3 figures and 1 tabl

Process for a Reactive Monomer Alignment Layer for Liquid Crystals Formed on an Azodye Sublayer

In this work, the detailed studies of surface polymerization stabilizing liquid crystal formed on an azodye sublayer are presented. The surface localized stabilization is obtained by free-radical polymerization of a dilute solution of a bi-functional reactive monomer (RM) in a liquid crystal (LC) solvent. To optimize the process for surface localized stabilization, we investigate the effects of several process parameters including RM concentration in LC hosts, the types of materials (either RM or LC), the photo-initiator (PI) concentration, ultra-violet (UV) polymerization intensity, and the UV curing temperature. The quality of surface localized stabilization is characterized and/or evaluated by optical microscopy, electro-optical behavior (transmission/voltage curve), the life test, and photo-bleaching. Our results show that, by carefully selecting materials, formulating mixtures, and controlling the polymerizing variables, the RM polymerization can be realized either at the surface or through the bulk. Overall, the combination of surface localized stabilization and photo-alignment offers an elegant and dynamic solution for controlling the alignment for LC, which could play a profound role in almost all liquid crystal optical devices. Keywords: photoalignment; liquid crystals; reactive monomers; azo dyeUnited States. Army Research Office (Contract W911NF-14-1-0650

The Droplet State and the Compressibility Anomaly in Dilute 2D Electron Systems

We investigate the space distribution of carrier density and the compressibility of two-dimensional (2D) electron systems by using the local density approximation. The strong correlation is simulated by the local exchange and correlation energies. A slowly varied disorder potential is applied to simulate the disorder effect. We show that the compressibility anomaly observed in 2D systems which accompanies the metal-insulator transition can be attributed to the formation of the droplet state due to disorder effect at low carrier densities.Comment: 4 pages, 3 figure