Subadditivity of Matrix phi-Entropy and Concentration of Random Matrices

Matrix concentration inequalities provide a direct way to bound the typical spectral norm of a random matrix. The methods for establishing these results often parallel classical arguments, such as the Laplace transform method. This work develops a matrix extension of the entropy method, and it applies these ideas to obtain some matrix concentration inequalities.Comment: 23 page

Properties of TiO2 thin films and a study of the TiO2-GaAs interface

Titanium dioxide (TiO2) films prepared by chemical vapor deposition were investigated in this study for the purpose of the application in the GaAs metal-insulator-semiconductor field-effect transistor. The degree of crystallization increases with the deposition temperature. The current-voltage study, utilizing an Al-TiO2-Al MIM structure, reveals that the d-c conduction through the TiO2 film is dominated by the bulk-limited Poole-Frenkel emission mechanism. The dependence of the resistivity of the TiO2 films on the deposition environment is also shown. The results of the capacitance-voltage study indicate that an inversion layer in an n-type substrate can be achieved in the MIS capacitor if the TiO2 films are deposited at a temperature higher than 275 C. A process of low temperature deposition followed by the pattern definition and a higher temperature annealing is suggested for device fabrications. A model, based on the assumption that the surface state densities are continuously distributed in energy within the forbidden band gap, is proposed to interpret the lack of an inversion layer in the Al-TiO2-GaAs MIS structure with the TiO2 films deposited at 200 C

The Masked Sample Covariance Estimator: An Analysis via Matrix Concentration Inequalities

Covariance estimation becomes challenging in the regime where the number p of variables outstrips the number n of samples available to construct the estimate. One way to circumvent this problem is to assume that the covariance matrix is nearly sparse and to focus on estimating only the significant entries. To analyze this approach, Levina and Vershynin (2011) introduce a formalism called masked covariance estimation, where each entry of the sample covariance estimator is reweighted to reflect an a priori assessment of its importance. This paper provides a short analysis of the masked sample covariance estimator by means of a matrix concentration inequality. The main result applies to general distributions with at least four moments. Specialized to the case of a Gaussian distribution, the theory offers qualitative improvements over earlier work. For example, the new results show that n = O(B log^2 p) samples suffice to estimate a banded covariance matrix with bandwidth B up to a relative spectral-norm error, in contrast to the sample complexity n = O(B log^5 p) obtained by Levina and Vershynin