The reconstruction of gene regulatory networks from gene expression data has been the subject of intense research activity. A variety of models and methods have been developed to address different aspects of this important problem. However, these techniques are narrowly focused on particular biological and experimental platforms, and require experimental data that are typically unavailable and difficult to ascertain. The more recent availability of higher-throughput sequencing platforms, combined with more precise modes of genetic perturbation, presents an opportunity to formulate more robust and comprehensive approaches to gene network inference. Here, we propose a step-wise framework for identifying gene-gene regulatory interactions that expand from a known point of genetic or chemical perturbation using time series gene expression data. This novel approach sequentially identifies non-steady state genes post-perturbation and incorporates them into a growing series of low-complexity optimization problems. The governing ordinary differential equations of this model are rooted in the biophysics of stochastic molecular events that underlie gene regulation, delineating roles for both protein and RNA-mediated gene regulation. We show the successful application of our core algorithms for network inference using simulated and real datasets.

We study the problem of high-dimensional covariance matrix estimation from partial observations. We consider covariance matrices modeled as Kronecker products of matrix factors, and rely on observations with missing values. In the absence of missing data, observation vectors are assumed to be i.i.d multivariate Gaussian. In particular, we propose a new procedure computationally affordable in high dimension to extend an existing permuted rank-penalized least-squares method to the case of missing data. Our approach is applicable to a large variety of missing data mechanisms, whether the process generating missing values is random or not, and does not require imputation techniques. We introduce a novel unbiased estimator and characterize its convergence rate to the true covariance matrix measured by the spectral norm of a permutation operator. We establish a tight outer bound on the square error of our estimate, and elucidate consequences of missing values on the estimation performance. Different schemes are compared by numerical simulations in order to test our proposed estimator.

We first consider a single hop wireless X network with K transmitters and J receivers, all with a single antenna. Each transmitter conveys an independent message for each receiver. The channel is assumed to have constant coefficients. We develop an interference alignment scheme for this setup and derive an achievable degrees of freedom (DoF) region. We show that in some cases, the derived region meets a previous outer bound, and hence, is the (exact) DoF region. For our achievability schemes, we divide each message into streams and use real interference alignment on the streams. Several previous results on the DoF region and total DoF for several special cases can be recovered from our result. Next, we consider a network model, such that each transmitter emits an arbitrary number of messages and each receiver can request an arbitrary subset of all emitted messages. We term this network the X network with multicast. We derive an achievability result for the DoF region of such networks, as well as an outer bound. Our achievability scheme is based on a novel extension of real interference alignment that enables us to achieve more points of the DoF region. This approach allows multiple messages from a single user to be aligned with other users' messages. Finally, we discuss some points on the outer bounds that are not achievable with any of the alignment schemes that we present.

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 Njantennas 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 (K, [N], [N]) interference channel with K users and N antennas at each node 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 setup 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.

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.

In this study, a two-user discrete memoryless multiple-access channel with two correlated states known non-causally at the corresponding encoder is considered. Each encoder transmits a message independent of the other encoder's and tries to cooperate with it. To consider cooperative encoders, it is assumed that each encoder strictly-causally receives and finds out the other encoder's transmitted signals and tries to cooperate with the other encoder by transmitting its message. A special case is also studied in which the common part of the states is known at the both encoders, resulting in a larger rate region. For these two scenarios, achievable rate regions are derived using a combination of block-Markov and Gel'fand-Pinsker coding techniques, and outer bounds are also established. For the second scenario, the achievable rate region of the Gaussian model is derived, and some implications of the results are discussed.

In this paper, we present a new method for variable elimination in systems of inequalities which is much faster than the Fourier-Motzkin Elimination (FME) method. In our method, a linear Diophantine problem is introduced which is dual to the original problem. The new Diophantine system is then solved, and the final result is calculated by finding the dual system of inequalities. This new method uses the algorithm Normaliz to find the Hilbert basis of the solution space of the given Diophantine problem. We introduce a problem in the interference channel with multiple nodes and solve it with this new method. Next, we generalize our method to all problems involving FME and compare the method with the previous method. Our method has many advantages in comparison to the previous method. It does not produce many of the redundant answers of the FME method. It also solves the whole problem in one step whereas the previous method uses a step by step approach in eliminating each auxiliary variable.

In this paper, a two-user discrete memoryless multiple-access channel (DM-MAC) with correlated channel states, each known at one of the encoders is considered, in which each encoder transmits independent messages and tries to cooperate with the other one. To consider cooperating encoders, it is assumed that each encoder strictly-causally receives and learns the other encoder's transmitted symbols and tries to cooperate with the other encoder by transmitting its message. Next, we study this channel in a special case; we assume that the common part of both states is known at both, hence encoders use this opportunity to get better rate region. For these scenarios, an achievable rate region is derived based on a combination of block-Markov encoding and Gel'fand-Pinsker coding techniques. Furthermore, the achievable rate region is established for the Gaussian channel, and it is shown that the capacity region is achieved in certain circumstances.