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This paper considers the portfolio problem for high dimensional data when the dimension and size are both large. We analyze the traditional Markowitz mean-variance (MV) portfolio by large dimension matrix theory, and find the spectral distribution of the sample covariance is the main factor to make the expected return of the traditional MV portfolio overestimate the theoretical MV portfolio. A correction is suggested to the spectral construction of the sample covariance to be the sample spectrally corrected covariance, and to improve the traditional MV portfolio to be spectrally corrected. In the expressions of the expected return and risk on the MV portfolio, the population covariance matrix is always a quadratic form, which will direct MV portfolio estimation. We provide the limiting behavior of the quadratic form with the sample spectrally-corrected covariance matrix, and explain the superior performance to the sample covariance as the dimension increases to infinity proportionally with the sample size. Moreover, this paper deduces the limiting behavior of the expected return and risk on the spectrally-corrected MV portfolio, and illustrates the superior properties of the spectrally-corrected MV portfolio. In simulations, we compare the spectrally-corrected estimates with the traditional and bootstrap-corrected estimates, and show the performance of the spectrally-corrected estimates are the best in portfolio returns and portfolio risk. We also compare the performance of the new proposed estimation with deferent optimal portfolio estimates for real data from S&P 500. The empirical findings are consistent with the theory developed in the paper.

This paper investigates the central limit theorem for linear spectral statistics of high dimensional sample covariance matrices of the form $${\mathbf {B}}_n=n^{-1}\sum _{j=1}^{n}{\mathbf {Q}}{\mathbf {x}}_j{\mathbf {x}}_j^{*}{\mathbf {Q}}^{*}$$ under the assumption that $$p/n\rightarrow y>0$$ , where $${\mathbf {Q}}$$ is a $$p\times k$$ nonrandom matrix and $$\{{\mathbf {x}}_j\}_{j=1}^n$$ is a sequence of independent k-dimensional random vector with independent entries. A key novelty here is that the dimension $$k\ge p$$ can be arbitrary, possibly infinity. This new model of sample covariance matrix $${\mathbf {B}}_n$$ covers most of the known models as its special cases. For example, standard sample covariance matrices are obtained with $$k=p$$ and $${\mathbf {Q}}={\mathbf {T}}_n^{1/2}$$ for some positive definite Hermitian matrix $${\mathbf {T}}_n$$ . Also with $$k=\infty $$ our model covers the case of repeated linear processes considered in recent high-dimensional time series literature. The CLT found in this paper substantially generalizes the seminal CLT in Bai and Silverstein (Ann Probab 32(1):553–605, 2004). Applications of this new CLT are proposed for testing the AR(1) or AR(2) structure for a causal process. Our proposed tests are then used to analyze a large fMRI data set.

The difficulty to efficiently compute the null distribution of the largest eigenvalue of a MANOVA matrix has hindered the wider applicability of Roy’s Largest Root Test (RLRT) though it was propose...

In this paper an ultra-low power, integrated pulse width modulated (PWM) temperature sensor is presented. The PWM temperature output is generated by the comparing a temperature dependent voltage with another voltage which is temperature independent. The proposed sensor leverages a 2-transistor voltage reference cell that is corrected using feedback through the bulk terminal to provide steady output across temperature. The feedback allows the reference to operate with supply voltages as low as 450 mV. Owing to the low supply voltage and operation in the sub-threshold region, the sensor has a measured power consumption as low as 47.2 nW (@ 27 °C) and 17.6 nW (@ −20 °C). Because of the low voltage required, it is suitable to be powered by harvested energy sources. To realize a robust temperature measurement, the temperature is measured using a ratio of output pulse-widths. A fabricated prototype in 65 nm CMOS has an area of $305\times 220 ~\mu \text{m}^{2}$ . The measured maximum temperature error is +1/−2.5 °C, and requires only a two-point calibration for operation over the temperature range −20–80 °C.

Transport-of-intensity diffraction tomography (TIDT) is a recently developed label-free computational microscopy technique that retrieves high-resolution three-dimensional (3D) refractive index (RI) distribution of biological specimens from 3D intensity-only measurements. However, the non-interferometric synthetic aperture in TIDT is generally achieved sequentially through the acquisition of a large number of through-focus intensity stacks captured at different illumination angles, resulting in a very cumbersome and redundant data acquisition process. To this end, we present a parallel implementation of a synthetic aperture in TIDT (PSA-TIDT) with annular illumination. We found that the matched annular illumination provides a mirror-symmetric 3D optical transfer function, indicating the analyticity in the upper half-plane of the complex phase function, which allows for recovery of the 3D RI from a single intensity stack. We experimentally validated PSA-TIDT by conducting high-resolution tomographic imaging of various unlabeled biological samples, including human breast cancer cell lines (MCF-7), human hepatocyte carcinoma cell lines (HepG2), Henrietta Lacks (HeLa) cells, and red blood cells (RBCs).

Summary We introduce a conceptually simple, efficient and easily implemented approach for learning the block structure in a large matrix. Using the properties of U-statistics and large-dimensional random matrix theory, the group structure of many variables can be directly identified based on the eigenvalues and eigenvectors of the scaled sample matrix. We also establish the asymptotic properties of the proposed approach under mild conditions. The finite-sample performance of the approach is examined by extensive simulations and data examples.

We present a new label-free three-dimensional (3D) microscopy technique, termed transport of intensity diffraction tomography with non-interferometric synthetic aperture (TIDT-NSA). Without resorting to interferometric detection, TIDT-NSA retrieves the 3D refractive index (RI) distribution of biological specimens from 3D intensity-only measurements at various illumination angles based on an off-the-shelf bright-field microscope with a programmable light-emitting-diode (LED) illumination source, allowing incoherent-diffraction-limited quantitative 3D phase-contrast imaging.

In this paper, a ridgelized Hotelling’s [Formula: see text] test is developed for a hypothesis on a large-dimensional mean vector under certain moment conditions. It generalizes the main result of Chen et al. [A regularized Hotelling’s [Formula: see text] test for pathway analysis in proteomic studies, J. Am. Stat. Assoc. 106(496) (2011) 1345–1360.] by relaxing their Gaussian assumption. This is achieved by establishing an exact four-moment theorem that is a simplified version of Tao and Vu’s [Random matrices: universality of local statistics of eigenvalues, Ann. Probab. 40(3) (2012) 1285–1315] work. Simulation results demonstrate the superiority of the proposed test over the traditional Hotelling’s [Formula: see text] test and its several extensions in high-dimensional situations.

The goal of this study was to test the equality of two covariance matrices by using modified Pillai’s trace statistics under a high-dimensional framework, i.e., the dimension and sample sizes go to infinity proportionally. In this paper, we introduce two modified Pillai’s trace statistics and obtain their asymptotic distributions under the null hypothesis. The benefits of the proposed statistics include the following: (1) the sample size can be smaller than the dimensions; (2) the limiting distributions of the proposed statistics are universal; and (3) we do not restrict the structure of the population covariance matrices. The theoretical results are established under mild and practical assumptions, and their properties are demonstrated numerically by simulations and a real data analysis.

This article presents a digitally frequency-tunable switched-capacitor power amplifier (SCPA) in 65-nm CMOS. The operable frequency range spans from 1.4 to 2.5 GHz. The center frequency is tuned by a programmable capacitor array, widely available from both monolithic microwave integrated circuit (MMIC) and MEMS manufacturers. A prototype is able to achieve a peak output power of 21.7 dBm with a peak system efficiency (SE) of 38.8%, while achieving 58.8% fractional bandwidth. When transmitting a 20-MHz, 64-quadrature amplitude modulation (QAM) long-term evolution (LTE) signal, it outputs 14.1 dBm at 15.7% SE with a measured adjacent channel leakage ratio (ACLR)

This is a complementary proof of partial generalized 4 moment theorem (PG4MT) mentioned and described in “Generalized Four Moment Theorem (G4MT) and its Application to CLT for Spiked Eigenvalues of High-dimensional Covariance Matrices”. Since the G4MT proposed in that paper requires both the matrices X and Y satisfying the assumption maxt,s|uts|2E{|x11|4I(|x11

We present a 3D label-free refractive index (RI) imaging technique based on single-exposure intensity diffraction tomography (sIDT) utilizing a color-multiplexed illumination scheme. In our method, the chromatic LEDs corresponding R/G/B channels in an annular programmable array provide oblique illumination geometry which matches the numerical aperture of the objective precisely to maximize the spectrum coverage. A color intensity image encoding the scattering field of the specimen from different directions is captured, and monochromatic intensity images with respect to three color channels were separated and then used to recover the 3D RI distribution of the object following the process of IDT. In addition, the axial chromatic dispersion of focal lengths at different wavelengths introduced by the chromatic aberration of the objective lens and the spatial position misalignment of the ring LED source in the imaging system’s transfer functions modeling are both corrected to significantly reduce the artifacts in slice-based deconvolution procedure for the reconstruction of 3D RI distribution. Experimental results on MCF-7, Spirulina algae, and live C. elegans samples demonstrate the solid performance of the sIDT method in label-free, high-throughput, and real-time (∼ 24 fps) 3D volumetric biological imaging applications.

Optical diffraction tomography (ODT) is a promising label-free three-dimensional (3D) microscopic method capable of measuring the 3D refractive index (RI) distribution of optically transparent samples (e.g., unlabeled biological cells). In recent years, non-interferometric ODT techniques have received increasing attention for their system simplicity, speckle-free imaging quality, and compatibility with existing microscopes. However, ODT methods for implementing non-interferometric measurements in high numerical aperture (NA) microscopy systems are often plagued by low-frequency missing problems—a consequence of violating the matched illumination condition. Here, we present transport-of-intensity Fourier ptychographic diffraction tomography (TI-FPDT) to address this challenging issue by combining ptychographic angular diversity with additional “transport of intensity” measurements. TI-FPDT exploits the defocused phase contrast to circumvent the stringent requirement on the illumination NA imposed by the matched illumination condition. It effectively overcomes the reconstruction quality deterioration and RI underestimation problems in conventional FPDT, as demonstrated by high-resolution tomographic imaging of various unlabeled transparent samples (including microspheres, USAF targets, HeLa cells, and C2C12 cells). Due to its simplicity and effectiveness, TI-FPDT is anticipated to open new possibilities for label-free 3D microscopy in various biomedical applications.

Large-dimensional (LD) random matrix theory, RMT for short, which originates from the research field of quantum physics, has shown tremendous capability in providing deep insights into large-dimensional systems. With the fact that we have entered an unprecedented era full of massive amounts of data and large complex systems, RMT is expected to play more important roles in the analysis and design of modern systems. In this paper, we review the key results of RMT and its applications in two emerging fields: wireless communications and deep learning. In wireless communications, we show that RMT can be exploited to design the spectrum sensing algorithms for cognitive radio systems and to perform the design and asymptotic analysis for large communication systems. In deep learning, RMT can be utilized to analyze the Hessian, input–output Jacobian and data covariance matrix of the deep neural networks, thereby to understand and improve the convergence and the learning speed of the neural networks. Finally, we highlight some challenges and opportunities in applying RMT to the practical large-dimensional systems.