For beta > 0, the beta-ensemble corresponds to the joint probability density on the real line proportional to prod_{n > m}^N abs{x_n - x_m}^beta prod_{n = 1}^N w(x_n) where w is the weight of the system. It has the application of being the Boltzmann factor for the configuration of N charge-one particles interacting logarithmically on an infinite wire inside an external field Q = -log w at inverse temperature beta. Similarly, the circular beta-ensemble has joint probability density proportional to prod_{n > m}^N abs{e^{itheta_n} - e^{itheta_m}}^beta prod_{n = 1}^N w(x_n) quad for theta_n in [- pi, pi) and can be interpreted as N charge-one particles on the unit circle interacting logarithmically with no external field. When beta = 1, 2, and 4, both ensembles are said to be solvable in that their correlation functions can be expressed in a form which allows for asymptotic calculations. It is not known, however, whether the general beta-ensemble is solvable. We present four families of particle models which are solvable point processes related to the beta-ensemble. Two of the examples interpolate between the circular beta-ensembles for beta = 1, 2, and 4. These give alternate ways of connecting the classical beta-ensembles besides simply changing the values of beta. The other two examples are "mirrored" particle models, where each particle has a paired particle reflected about some point or axis of symmetry.

In this thesis, we focus on studying the performance of general regularized discriminant analysis (RDA) classifiers. The data used for analysis is assumed to follow Gaussian mixture model with different means and covariances. RDA offers a rich class of regularization options, covering as special cases the regularized linear discriminant analysis (RLDA) and the regularized quadratic discriminant analysis (RQDA) classi ers. We analyze RDA under the double asymptotic regime where the data dimension and the training size both increase in a proportional way. This double asymptotic regime allows for application of fundamental results from random matrix theory. Under the double asymptotic regime and some mild assumptions, we show that the asymptotic classification error converges to a deterministic quantity that only depends on the data statistical parameters and dimensions. This result not only implicates some mathematical relations between the misclassification error and the class statistics, but also can be leveraged to select the optimal parameters that minimize the classification error, thus yielding the optimal classifier. Validation results on the synthetic data show a good accuracy of our theoretical findings. We also construct a general consistent estimator to approximate the true classification error in consideration of the unknown previous statistics. We benchmark the performance of our proposed consistent estimator against classical estimator on synthetic data. The observations demonstrate that the general estimator outperforms others in terms of mean squared error (MSE).

The covariance estimation is one of the most critical tasks in multivariate statistical analysis. In many applications, reliable estimation of the covariance matrix, or scatter matrix in general, is required. The performance of the classical maximum likelihood method relies a great deal on the validity of the model assumption. Since the assumptions are often approximately correct, many robust statistical methods have been proposed to be robust against the deviation from the model assumptions. M-estimator is an important class of robust estimator of the scatter matrix. The properties of these robust estimators under high dimensional setting, which means the number of dimensions has the same order of magnitude as the number of observations, is desirable. To study these, random matrix theory is a very important tool. With high dimensional properties of robust estimators, we introduced a new method for blind spectrum sensing in cognitive radio networks.

This work utilizes zero-beta CAPM to derive an alternative form dubbed the ZCAPM. The ZCAPM posits that asset prices are a function of market risk composed of two components: average market returns and cross-sectional market volatility. Market risk associated with average market returns in the CAPM market model is known as beta risk. We refer to market risk related to cross-sectional market volatility as zeta risk. Using U.S. stock returns from January 1965 to December 2010, out-of-sample cross-sectional asset pricing tests show that the ZCAPM better predicts stock returns than popular three- and four-factor models. These and other empirical tests lead us to conclude that the ZCAPM holds promise as a robust asset pricing model.

The biological datasets produced as a result of high-throughput genomic research such as specifically microarrays, contain vast amounts of knowledge for entire genome and their expression affiliations. Gene clustering from such data is a challenging task due to the huge data size and high complexity of the algorithms as well as the visualization needs. Most of the existing analysis methods for genome-wide gene expression profiles are sequential programs using greedy algorithms and require subjective human decision. Recently, Zhu et al. proposed a parallel Random matrix theory (RMT) based approach for generating transcriptional networks, which is much more resistant to high level of noise in the data [9] without human intervention. Nowadays GPUs are designed to be used more efficiently for general purpose computing [1] and are vastly superior to CPUs [6] in terms of threading performance. Our kernel functions running on GPU utilizes the functions from both the libraries of Compute Unified Basic Linear Algebra Subroutines (CUBLAS) and Compute Unified Linear Algebra (CULA) which implements the Linear Algebra Package (LAPACK). Our experiment results show that GPU program can achieve an average speed-up of 2~3 times for some simulated datasets.

Random matrix theory is an outstanding mathematical tool that has demonstrated its usefulness in many areas ranging from wireless communication to finance and economics. The main motivation behind its use comes from the fundamental role that random matrices play in modeling unknown and unpredictable physical quantities. In many situations, meaningful metrics expressed as scalar functionals of these random matrices arise naturally. Along this line, the present work consists in leveraging tools from random matrix theory in an attempt to answer fundamental questions related to applications from statistical signal processing and machine learning. In a first part, this thesis addresses the development of analytical tools for the computation of the inverse moments of random Gram matrices with one side correlation. Such a question is mainly driven by applications in signal processing and wireless communications wherein such matrices naturally arise. In particular, we derive closed-form expressions for the inverse moments and show that the obtained results can help approximate several performance metrics of common estimation techniques. Then, we carry out a large dimensional study of discriminant analysis classifiers. Under mild assumptions, we show that the asymptotic classification error approaches a deterministic quantity that depends only on the means and covariances associated with each class as well as the problem dimensions. Such result permits a better understanding of the underlying classifiers, in practical large but finite dimensions, and can be used to optimize the performance. Finally, we revisit kernel ridge regression and study a centered version of it that we call centered kernel ridge regression or CKRR in short. Relying on recent advances on the asymptotic properties of random kernel matrices, we carry out a large dimensional analysis of CKRR under the assumption that both the data dimesion and the training size grow simultaneiusly large at the same rate. We particularly show that both the empirical and prediction risks converge to a limiting risk that relates the performance to the data statistics and the parameters involved. Such a result is important as it permits a better undertanding of kernel ridge regression and allows to efficiently optimize the performance.

フランス国リヨン高等師範学校との共同学位プログラムによる学位 / 京都大学 / 新制・課程博士 / 博士(理学) / 甲第23449号 / 理博第4743号 / 新制||理||1680(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 COLLINS Benoit Vincent Pierre, 教授 泉 正己, 教授 日野 正訓 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

By endowing the base station with hundreds of antennas and relying on spatial multiplexing, massive multiple-input-multiple-output (MIMO) allows impressive advantages in many fronts. To reduce this promising technology to reality, thorough performance analysis has to be conducted. Along this line, this work is focused on the convenient high-dimensionality of massive MIMO’s corresponding model. Indeed, the large number of antennas allows us to harness asymptotic results from Random Matrix Theory to provide accurate approximations of the main performance metrics. The derivations yield simple closed-form expressions that can be easily interpreted and manipulated in contrast to their alternative random equivalents. Accordingly, in this dissertation, we investigate and optimize the performance of massive MIMO in different contexts. First, we explore the spectral efficiency of massive MIMO in large-scale multi-tier heterogeneous networks that aim at network densification. This latter is epitomized by the joint implementation of massive MIMO and small cells to reap their benefits. Our interest is on the design of coordinated beamforming that mitigates cross-tier interference. Thus, we propose a regularized SLNR-based precoding in which the regularization factor is used to allow better resilience to channel estimation errors. Second, we move to studying massive MIMO under Line-of-Sight (LoS) propagation conditions. To this end, we carry out an analysis of the uplink (UL) of a massive MIMO system with per-user channel correlation and Rician factor. We start by analyzing conventional processing schemes such as LMMSE and MRC under training-based imperfect-channel-estimates, and then, propose a statistical combining technique that is more suitable in LoS-prevailing environments. Finally, we look into the interplay between LoS and the fundamental limitation of massive MIMO systems, namely, pilot contamination. We propose to analyze and compare the performance using single-cell and multi-cell detection methods. In this regard, the single-cell schemes are shown to produce higher SEs as the LoS strengthens, yet remain hindered by LoS-induced interference and pilot contamination. In contrast, for multi-cell combining, we analytically demonstrate that M-MMSE outperforms both single-cell detectors by generating a capacity that scales linearly with the number of antennas, and is further enhanced with LoS.

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No / Eigenvalue based spectrum sensing schemes such as Maximum Minimum Eigenvalue (MME), Maximum Energy Detection (MED) and Energy with Minimum Eigenvalue (EME) have higher spectrum sensing performance without requiring any prior knowledge of Primary User (PU) signal but the decision hypothesis used in these eigenvalue based sensing schemes depends on the calculation of maximum eigenvalue from covariance matrix of measured signal. Calculation of the covariance matrix followed by eigenspace analysis of the covariance matrix is a resource intensive operation and takes overhead time during critical process of spectrum sensing. In this paper we propose a new blind spectrum sensing scheme based on the approximation of the maximum eigenvalue using state of the art results from Random Matrix Theory (RMT). The proposed sensing scheme has been evaluated through extensive simulations on wireless microphone signals and the proposed scheme shows higher probability of detection (Pd) performance. The proposed spectrum sensing also shows higher detection performance as compared to energy detection scheme and RMT based sensing schemes such as MME and EME.