In this work, we study and extend algorithms for Sparse Regression and Causal Inference problems. Both the problems are fundamental in the area of Data Science. The goal of regression problem is to nd out the \best" relationship between an output variable and input variables, given samples of the input and output values. We consider sparse regression under a high-dimensional linear model with strongly correlated variables, situations which cannot be handled well using many existing model selection algorithms. We study the performance of the popular feature selection algorithms such as LASSO, Elastic Net, BoLasso, Clustered Lasso as well as Projected Gradient Descent algorithms under this setting in terms of their running time, stability and consistency in recovering the true support. We also propose a new feature selection algorithm, BoPGD, which cluster the features rst based on their sample correlation and do subsequent sparse estimation using a bootstrapped variant of the projected gradient descent method with projection on the non-convex L0 ball. We attempt to characterize the efficiency and consistency of our algorithm by performing a host of experiments on both synthetic and real world datasets. Discovering causal relationships, beyond mere correlation, is widely recognized as a fundamental problem. The Causal Inference problems use observations to infer the underlying causal structure of the data generating process. The input to these problems is either a multivariate time series or i.i.d sequences and the output is a Feature Causal Graph where the nodes correspond to the variables and edges capture the direction of causality. For high dimensional datasets, determining the causal relationships becomes a challenging task because of the curse of dimensionality. Graphical modeling of temporal data based on the concept of \Granger Causality" has gained much attention in this context. The blend of Granger methods along with model selection techniques, such as LASSO, enables efficient discovery of a \sparse" sub-set of causal variables in high dimensional settings. However, these temporal causal methods use an input parameter, L, the maximum time lag. This parameter is the maximum gap in time between the occurrence of the output phenomenon and the causal input stimulus. How-ever, in many situations of interest, the maximum time lag is not known, and indeed, finding the range of causal e ects is an important problem. In this work, we propose and evaluate a data-driven and computationally efficient method for Granger causality inference in the Vector Auto Regressive (VAR) model without foreknowledge of the maximum time lag. We present two algorithms Lasso Granger++ and Group Lasso Granger++ which not only constructs the hypothesis feature causal graph, but also simultaneously estimates a value of maxlag (L) for each variable by balancing the trade-o between \goodness of t" and \model complexity".

It is well known that multiple antennas at the transmitter and receiver are imperative for reliable and high data-rate communication over wireless channels. However, these systems essentially need multiple radio frequency (RF) chains owing to multiple antennas, and hence pose challenges for applications with limited form-factor. Antenna Selection (AS) techniques alleviate this problem by using only a subset of the total available antennas and hence require only a few RF chains compared to the number of antennas. These systems operate in a closed-loop scenario, where the information fed back from the receiver is used for the transmit antenna subset selection. In contrast to this, a novel open-loop technique known as spatial modulation (SM) was recently proposed that uses a single RF-chain at the transmitter and achieves a higher spectral efficiency compared to single-input and AS based systems. The work in the thesis mainly focuses on the following aspects of SM system: Study of Mutual Information in SM systems operating in open-loop and closed-loop scenarios: We study the achievable mutual information in the SM system operating with finite and Gaussian input alphabet, and compare the results with that of the SIMO and AS based systems. Reduced-complexity maximum-likelihood (ML) decoding algorithms for SM systems: We propose ML-optimal sphere decoders for SM systems with arbitrary number of transmit antennas. Furthermore, a reduced-complexity ML detector is also proposed whose computational complexity is lowest among the known existing detectors in the literature. Transmit diversity techniques for SM systems: The conventional SM system achieves a transmit diversity order of one. We propose a complex interleaved orthogonal design baaed SM scheme that achieves a transit diversity order of two, while offering symbol-by- symbol ML decodability. Transmit antenna subset selection algorithms for SM systems: The SM system is considered in the closed-loop scenario, where only a subset of the total number of transmit antennas is chosen based on the information fed back by the receiver. Specifically, the Euclidean distance and capacity optimized antenna selection algorithms are studied in comparison with the conventional AS based systems. SM system operating in dispersive channels: The SM system operating in a dispersive channel with the aid of zero-padding is studied. It is shown that the SM system achieves full receive-diversity and multipath-diversity with ML decoding, but offers a decoding complexity that is exponential in the number of multipaths. Furthermore, a reduced complexity linear receiver is proposed that achieves achieves full multipath as well as receive-diversity, while offering a decoding complexity order same as that of the SM system operating in a frequency-flat channel.

Opportunistic selection algorithms have grown in importance as next generation wireless systems strive towards higher data rates and spectral efficiencies. For example, in orthogonal frequency division multiple access(OFDMA), the system bandwidth is divided into many sub channels. For each sub channel, the user with the highest channel gain is opportunistically assigned to it. .Likewise, in a multi-source, multi-destination (MSD) cooperative relay system, a relay node must be assigned for every source-destination (SD) pair. The assignment decisions are based on local channel knowledge and must be fast so as to maximize the time available for data transmission. We develop novel multiple access based splitting-based selection algorithms for OFDMA and MSD systems. These systems are unique in that the same user and relay can be the most suitable one for multiple sub channels and multiple SD pairs, respectively. For OFDMA systems, we propose an algorithm called Split Select that assigns for every sub channel the user with the highest channel gain over it. For MSD systems, we propose a contention-based en masse assignment (CBEA) algorithm that assigns to each SD pair a relay that is capable of aiding it. Both Split Select and CBEA are fast and scale well with the number of nodes. For example, Split Select requires just 2.2 slots, on average, to assign a sub channel to its best user even when there are an asymptotically large number of contending users. Likewise, CBEA often takes far less than one slot, on average, to assign a relay to each SD pair.