Multiple-Antenna Interference Channel with Receive Antenna Joint Processing and Real Interference Alignment

We consider a constant $K$-user Gaussian interference channel with $M$ antennas at each transmitter and $N$ antennas at each receiver, denoted as a $(K,M,N)$ channel. Relying on a result on simultaneous Diophantine approximation, a real interference alignment scheme with joint receive antenna processing is developed. The scheme is used to provide new proofs for two previously known results, namely 1) the total degrees of freedom (DoF) of a $(K, N, N)$ channel is $NK/2$; and 2) the total DoF of a $(K, M, N)$ channel is at least $KMN/(M+N)$. We also derive the DoF region of the $(K,N,N)$ channel, and an inner bound on the DoF region of the $(K,M,N)$ channel

Network topology identification based on measured data

We consider the problem of modeling of systems and learning of models from a limited number of measurements. We also contribute to the development of inference algorithms that require high-dimensional data processing. As an inspiring example, a growing interest in biology is to determine dependencies among genes. Such problem, known as gene regulatory network inference, often leads to identifying of large networks through relatively small gene expression data. The main purpose of the thesis is to develop models and learning methods for data based applications. In particular, we first build a dynamical model for gene-gene interactions to learn the topology of gene regulatory networks from gene expression data. Our proposed model is applicable to such complex gene regulatory networks that contain loops and non-linear dependencies between genes. We seek to use dynamical gene expression data when a system is perturbed. Ideally, such dynamical changes result from local genetic or chemical perturbations of systems in steady state that can be captured in a time-dependent manner. We present a low-complexity inference method that can be adapted to incorporate other information measured across a biological system. The performance of our method is examined employing both simulated and real datasets. This work can potentially inform biological discovery relating to interactions of genes in disease-relevant networks, synthetic networks, and networks immediate to drug response. Along with the main objective of the thesis, we next seek to estimate high-dimensional covariance matrices based on a few partial observations. Notably, covariance matrices can be utilized to form networks or improve network inference. We assume that the true covariance matrix can be modeled as a sum of Kronecker products of two lower dimensional matrices. To estimate covariance, we propose a convex optimization approach computationally affordable in high-dimensional setting and applicable to missing data. Regardless of whether the process producing missing values is random or not, our novel scheme can be used without employing any imputation methods. We characterize the symmetry and positive definiteness of the estimated covariance and further shed light on its square error performance. The effect of missing values on the estimation error is mathematically presented and numerical results are illustrated to validate our method. In addition to the modeling and learning, we improve inference algorithms that involve high-dimensional data processing. Specifically, we attempt to reduce the complexity of the linear minimum mean-square error (LMMSE) estimation when observation vectors have high-dimensionality and contain missing entries. In this context, the standard LMMSE estimator must be re-computed whenever missing values take place at different positions. Instead, we propose a method to first construct the LMMSE estimator based on complete data statistics. We then apply this estimator to the data vector with missing values replaced by zeros. We finally establish a low-complexity update according to missing data patterns to modify our estimation and preserve the LMMSE optimality

Multiple-Antenna Interference Channels with Real Interference Alignment and Receive Antenna Joint Processing

In this paper, the degrees of freedom (DoF) regions of constant coefficient multiple antenna interference channels are investigated. First, we consider a K-user Gaussian interference channel with Mk antennas at transmitter k, 1≤k≤K, and Nj antennas at receiver j, 1≤j≤K, denoted as a (K,[Mk],[Nj]) channel. Relying on a result of simultaneous Diophantine approximation, a real interference alignment scheme with joint receive antenna processing is developed. The scheme is used to obtain an achievable DoF region. The proposed DoF region includes two previously known results as special cases, namely 1) the total DoF of a K-user interference channel with N antennas at each node, (K,[N],[N]) channel, is NK/2; and 2) the total DoF of a (K,[M],[N]) channel is at least KMN/(M+N). We next explore constant-coefficient interference networks with K transmitters and J receivers, all having N antennas. Each transmitter emits an independent message and each receiver requests an arbitrary subset of the messages. Employing the novel joint receive antenna processing, the DoF region for this set-up is obtained. We finally consider wireless X networks where each node is allowed to have an arbitrary number of antennas. It is shown that the joint receive antenna processing can be used to establish an achievable DoF region, which is larger than what is possible with antenna splitting. As a special case of the derived achievable DoF region for constant coefficient X network, the total DoF of wireless X networks with the same number of antennas at all nodes and with joint antenna processing is tight while the best inner bound based on antenna splitting cannot meet the outer bound. Finally, we obtain a DoF region outer bound based on the technique of transmitter grouping

Diophantine Approximation and applications in Interference Alignment

This paper is motivated by recent applications of Diophantine approximation in electronics, in particular, in the rapidly developing area of Interference Alignment. Some remarkable advances in this area give substantial credit to the fundamental Khintchine-Groshev Theorem and, in particular, to its far reaching generalisation for submanifolds of a Euclidean space. With a view towards the aforementioned applications, here we introduce and prove quantitative explicit generalisations of the Khintchine-Groshev Theorem for non-degenerate submanifolds of R n. The importance of such quantitative statements is explicitly discussed in Jafar's monograph [12, §4.7.1]

Network topology identification based on measured data

We consider the problem of modeling of systems and learning of models from a limited number of measurements. We also contribute to the development of inference algorithms that require high-dimensional data processing. As an inspiring example, a growing interest in biology is to determine dependencies among genes. Such problem, known as gene regulatory network inference, often leads to identifying of large networks through relatively small gene expression data. The main purpose of the thesis is to develop models and learning methods for data based applications. In particular, we first build a dynamical model for gene-gene interactions to learn the topology of gene regulatory networks from gene expression data. Our proposed model is applicable to such complex gene regulatory networks that contain loops and non-linear dependencies between genes. We seek to use dynamical gene expression data when a system is perturbed. Ideally, such dynamical changes result from local genetic or chemical perturbations of systems in steady state that can be captured in a time-dependent manner. We present a low-complexity inference method that can be adapted to incorporate other information measured across a biological system. The performance of our method is examined employing both simulated and real datasets. This work can potentially inform biological discovery relating to interactions of genes in disease-relevant networks, synthetic networks, and networks immediate to drug response. Along with the main objective of the thesis, we next seek to estimate high-dimensional covariance matrices based on a few partial observations. Notably, covariance matrices can be utilized to form networks or improve network inference. We assume that the true covariance matrix can be modeled as a sum of Kronecker products of two lower dimensional matrices. To estimate covariance, we propose a convex optimization approach computationally affordable in high-dimensional setting and applicable to missing data. Regardless of whether the process producing missing values is random or not, our novel scheme can be used without employing any imputation methods. We characterize the symmetry and positive definiteness of the estimated covariance and further shed light on its square error performance. The effect of missing values on the estimation error is mathematically presented and numerical results are illustrated to validate our method. In addition to the modeling and learning, we improve inference algorithms that involve high-dimensional data processing. Specifically, we attempt to reduce the complexity of the linear minimum mean-square error (LMMSE) estimation when observation vectors have high-dimensionality and contain missing entries. In this context, the standard LMMSE estimator must be re-computed whenever missing values take place at different positions. Instead, we propose a method to first construct the LMMSE estimator based on complete data statistics. We then apply this estimator to the data vector with missing values replaced by zeros. We finally establish a low-complexity update according to missing data patterns to modify our estimation and preserve the LMMSE optimality.</p

Multiple-Antenna Interference Channels with Real Interference Alignment and Receive Antenna Joint Processing

In this paper, the degrees of freedom (DoF) regions of constant coefficient multiple antenna interference channels are investigated. First, we consider a K-user Gaussian interference channel with Mk antennas at transmitter k, 1≤k≤K, and Nj antennas at receiver j, 1≤j≤K, denoted as a (K,[Mk],[Nj]) channel. Relying on a result of simultaneous Diophantine approximation, a real interference alignment scheme with joint receive antenna processing is developed. The scheme is used to obtain an achievable DoF region. The proposed DoF region includes two previously known results as special cases, namely 1) the total DoF of a K-user interference channel with N antennas at each node, (K,[N],[N]) channel, is NK/2; and 2) the total DoF of a (K,[M],[N]) channel is at least KMN/(M+N). We next explore constant-coefficient interference networks with K transmitters and J receivers, all having N antennas. Each transmitter emits an independent message and each receiver requests an arbitrary subset of the messages. Employing the novel joint receive antenna processing, the DoF region for this set-up is obtained. We finally consider wireless X networks where each node is allowed to have an arbitrary number of antennas. It is shown that the joint receive antenna processing can be used to establish an achievable DoF region, which is larger than what is possible with antenna splitting. As a special case of the derived achievable DoF region for constant coefficient X network, the total DoF of wireless X networks with the same number of antennas at all nodes and with joint antenna processing is tight while the best inner bound based on antenna splitting cannot meet the outer bound. Finally, we obtain a DoF region outer bound based on the technique of transmitter grouping.This is a pre-print of the article Zamanighomi, Mahdi, and Zhengdao Wang. "Multiple-Antenna Interference Channels with Real Interference Alignment and Receive Antenna Joint Processing," arXiv: https://arxiv.org/abs/1304.4567v2 (2014).</p