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We study the performance of first- and second-order optimization methods for l_1-regularized sparse least-squares problems as the conditioning of the problem changes and the dimensions of the problem increase up to one trillion. A rigorously defined generator is presented which allows control of the dimensions, the conditioning and the sparsity of the problem. The generator has very low memory requirements and scales well with the dimensions of the problem.

Consider a regression model with fixed design and Gaussian noise where the regression function can potentially be well approximated by a function that admits a sparse representation in a given dictionary. This paper resorts to exponential weights to exploit this underlying sparsity by implementing the principle of sparsity pattern aggregation. This model selection take on sparse estimation allows us to derive sparsity oracle inequalities in several popular frameworks, including ordinary sparsity, fused sparsity and group sparsity. One striking aspect of these theoretical results is that they hold under no condition in the dictionary. Moreover, we describe an efficient implementation of the sparsity pattern aggregation principle that compares favorably to state-of-the-art procedures on some basic numerical examples.

Massive multiple input multiple output (MIMO) systems can significantly improve the channel capacity by deploying multiple antennas at the transmitter and receiver. Massive MIMO is considered as one of key technologies of the next generation of wireless communication systems. However, with the increase of the number of antennas at the base station, a large number of unknown channel parameters need to be dealt with, which makes the channel estimation a challenging problem. Hence, the research on the channel estimation for massive MIMO is of great importance to the development of the next generation of communication systems. The wireless multipath channel exhibits sparse characteristics, but the traditional channel estimation techniques do not make use of the sparsity. The channel estimation based on compressive sensing (CS) can make full use of the channel sparsity, while use fewer pilot symbols. In this work, CS channel estimation methods are proposed for massive MIMO systems in complex environments operating in multipath channels with static and time-varying parameters. Firstly, a CS channel estimation algorithm for massive MIMO systems with Orthogonal Frequency Division Multiplexing (OFDM) is proposed. By exploiting the spatially common sparsity in the virtual angular domain of the massive MIMO channels, a dichotomous-coordinate-decent-joint-sparse-recovery (DCD-JSR) algorithm is proposed. More specifically, by considering the channel is static over several OFDM symbols and exhibits common sparsity in the virtual angular domain, the DCD-JSR algorithm can jointly estimate multiple sparse channels with low computational complexity. The simulation results have shown that, compared to existing channel estimation algorithms such as the distributed-sparsity-adaptive-matching-pursuit (DSAMP) algorithm, the proposed DCD-JSR algorithm has significantly lower computational complexity and better performance. Secondly, these results have been extended to the case of multipath channels with time-varying parameters. This has been achieved by employing the basis expansion model to approximate the time variation of the channel, thus the modified DCD-JSR algorithm can estimate the channel in a massive MIMO OFDM system operating over frequency selective and highly mobile wireless channels. Simulation results have shown that, compared to the DCD-JSR algorithm designed for time-invariant channels, the modified DCD-JSR algorithm provides significantly better estimation performance in fast time-varying channels.

This paper treats the problem of minimizing a general continuously differentiable function subject to sparsity constraints. We present and analyze several different optimality criteria which are based on the notions of stationarity and coordinatewise optimality. These conditions are then used to derive three numerical algorithms aimed at finding points satisfying the resulting optimality criteria: the iterative hard thresholding method and the greedy and partial sparse-simplex methods. The first algorithm is essentially a gradient projection method, while the remaining two algorithms are of a coordinate descent type. The theoretical convergence of these techniques and their relations to the derived optimality conditions are studied. The algorithms and results are illustrated by several numerical examples. [PUBLICATION ABSTRACT]

We add a set of convex constraints to the lasso to produce sparse interaction models that honor the hierarchy restriction that an interaction only be included in a model if one or both variables are marginally important. We give a precise characterization of the effect of this hierarchy constraint, prove that hierarchy holds with probability one and derive an unbiased estimate for the degrees of freedom of our estimator. A bound on this estimate reveals the amount of fitting \"saved\" by the hierarchy constraint. We distinguish between parameter sparsity—the number of nonzero coefficients—and practical sparsity—the number of raw variables one must measure to make a new prediction. Hierarchy focuses on the latter, which is more closely tied to important data collection concerns such as cost, time and effort. We develop an algorithm, available in the R package hierNet, and perform an empirical study of our method.

This thesis proposes a parsimonious approach to localization, mapping and object recognition for a pseudo-mobile robot equipped with a biomimetic array of tactile whiskers to autonomously interact, explore and represent a real-world environment. Tactile whisker sensors enable the robotic platform to perceive unique environmental properties and can operate in extreme conditions that preclude the use of conventional sensors, however, such sensors are disadvantaged by their limited range and sample sparsity. To address the sparsity, the information contained in each contact should be fully exploited, whilst the limited range of the array can be addressed through appropriate movement and placement of the whiskers and the array. An existing Simultaneous Localization and Mapping (SLAM) algorithm called RatSLAM was adopted as the basis for the inference of location and demonstrated as suitable for correcting odometry errors using whisker tactile sensing. The adoption of a closed loop contact induced whisker placement strategy, directly inspired by rat whisking behavior, improved the performance of the algorithm in further reducing odometry error. The fidelity of object shape reconstruction through the forward kinematic projection of whisker contact locations was analyzed and a number of machine learning approaches compared to assess their ecacy at discerning radial distance to contact and thus improve object shape reconstruction. A support vector regression technique was found to reliably improve estimates of radial distance to contact along the whisker shaft following natural, unconstrained whisker contacts. A framework for combining the 3D pose estimation from RatSLAM with a 6D pose estimation system suitable for object recognition is proposed with the 6D system implemented and demonstrated correctly identifying household objects through tactile whisker exploration. The adoption of whisker array placement strategies inspired by cutaneous-tactile research improved the robustness of object identification and two regional search strategies were investigated for the purpose of reducing the time taken to correctly classify objects.

The optimization models with sparsity arise in many areas of science and engineering, such as compressive sensing, image processing, statistical learning and machine learning. In this thesis, we study a general 10-minimization model, which can be used to deal with many practical applications. Firstly, we show some theoretical properties of the solutions of this model. Then, two types of re-weighted 11-algorithms will be developed from both the perspectives of primal and dual spaces, respectively. The primal re-weighted 11-algorithms will be derived through the 1st-order approximation of the so-called merit functions for sparsity. The dual re-weighted 11-algorithms for the general 10-model will be developed based on the reformulation of the general 10-model as a certain bilevel programming problem under the assumption of strict complementarity. We conduct numerical experiments to demonstrate the efficiency of the primal and dual re-weighted 11-algorithms and compare with some existing algorithms. We also establish a general stability result for a class of 11-minimization approach which is broad enough to cover many important special cases. Unlike the existing stability results developed under the null space property and restricted isotonic property, we use a classic Hoffman's theorem to establish a restricted-weak-RSP-based stability result for this class of 11-minimization approach.

We analyze the performance of spectral clustering for community extraction in stochastic block models. We show that, under mild conditions, spectral clustering applied to the adjacency matrix of the network can consistently recover hidden communities even when the order of the maximum expected degree is as small as log n, with n the number of nodes. This result applies to some popular polynomial time spectral clustering algorithms and is further extended to degree corrected stochastic block models using a spherical k-median spectral clustering method. A key component of our analysis is a combinatorial bound on the spectrum of binary random matrices, which is sharper than the conventional matrix Bernstein inequality and may be of independent interest.

Issue Title: Special Issue on Supercomputing

In this paper, we study the sparsity-adapted complex moment-Hermitian sum of squares (moment-HSOS) hierarchy for complex polynomial optimization problems, where the sparsity includes correlative sparsity and term sparsity. We compare the strengths of the sparsity-adapted complex moment-HSOS hierarchy with the sparsity-adapted real moment-SOS hierarchy on either randomly generated complex polynomial optimization problems or the AC optimal power flow problem. The results of numerical experiments show that the sparsity-adapted complex moment-HSOS hierarchy provides a trade-off between the computational cost and the quality of obtained bounds for large-scale complex polynomial optimization problems.