Non-convex Estimator Research Articles

High-dimensional low-rank tensor autoregressive time series modeling

Modern technological advances have enabled an unprecedented amount of structured data with complex temporal dependence, urging the need for new methods to efficiently model and forecast high-dimensional tensor-valued time series. This paper provides a new modeling framework to accomplish this task via autoregression (AR). By considering a low-rank Tucker decomposition for the transition tensor, the proposed tensor AR can flexibly capture the underlying low-dimensional tensor dynamics, providing both substantial dimension reduction and meaningful multi-dimensional dynamic factor interpretations. For this model, we first study several nuclear-norm-regularized estimation methods and derive their non-asymptotic properties under the approximate low-rank setting. In particular, by leveraging the special balanced structure of the transition tensor, a novel convex regularization approach based on the sum of nuclear norms of square matricizations is proposed to efficiently encourage low-rankness of the coefficient tensor. To further improve the estimation efficiency under exact low-rankness, a non-convex estimator is proposed with a gradient descent algorithm, and its computational and statistical convergence guarantees are established. Simulation studies and an empirical analysis of tensor-valued time series data from multi-category import-export networks demonstrate the advantages of the proposed approach.

Uncertainty Quantification for Nonconvex Tensor Completion: Confidence Intervals, Heteroscedasticity and Optimality

We study the distribution and uncertainty of nonconvex optimization for noisy tensor completion—the problem of estimating a low-rank tensor given incomplete and corrupted observations of its entries. Focusing on a two-stage estimation algorithm proposed by Cai <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> , we characterize the distribution of this nonconvex estimator down to fine scales. This distributional theory in turn allows one to construct valid and short confidence intervals for both the unseen tensor entries and the unknown tensor factors. The proposed inferential procedure enjoys several important features: (1) it is fully adaptive to noise heteroscedasticity, and (2) it is data-driven and automatically adapts to unknown noise distributions. Furthermore, our findings unveil the statistical optimality of nonconvex tensor completion: it attains un-improvable <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula> accuracy—including both the rates and the pre-constants—when estimating both the unknown tensor and the underlying tensor factors.

Target Localization in Wireless Sensor Networks Based on Received Signal Strength and Convex Relaxation.

A new positioning algorithm based on RSS measurement is proposed. The algorithm adopts maximum likelihood estimation and semi-definite programming. The received signal strength model is transformed to a non-convex estimator for the positioning of the target using the maximum likelihood estimation. The non-convex estimator is then transformed into a convex estimator by semi-definite programming, and the global minimum of the target location estimation is obtained. This algorithm aims at the known problem and then extends its application to the case of unknown. The simulations and experimental results show that the proposed algorithm has better accuracy than the existing positioning algorithms.

An SOCP Estimator for Hybrid RSS and AOA Target Localization in Sensor Networks.

This work addresses the problem of target localization in three-dimensional wireless sensor networks (WSNs). The proposed algorithm is based on a hybrid system that employs angle of arrival (AOA) and received signal strength (RSS) measurements, where the target’s transmit power is considered as an unknown parameter. Although both cases of a known and unknown target’s transmit power have been addressed in the literature, most of the existing approaches for unknown transmit power are either carried out recursively, or require a high computational cost. This results in an increased execution time of these algorithms, which we avoid in this work by proposing a single-iteration solution with moderate computational complexity. By exploiting the measurement models, a non-convex least squares (LS) estimator is derived first. Then, to tackle its nonconvexity, we resort to second-order cone programming (SOCP) relaxation techniques to transform the non-convex estimator into a convex one. Additionally, to make the estimator tighter, we exploit the angle between two vectors by using the definition of their inner product, which arises naturally from the derivation steps that are taken. The proposed method not only matches the performance of a computationally more complex state-of-the-art method, but it outperforms it for small N. This result is of a significant value in practice, since one desires to localize the target using the least number of anchor nodes as possible due to network costs.

3-D RSS-AOA Based Target Localization Method in Wireless Sensor Networks Using Convex Relaxation

This paper addresses a target localization problem in 3-D wireless sensor networks using a hybrid system that fuses received signal strength and angle of arrival measurements. First, we formulate the received signal strength and angle of arrival measurement models as the pseudo-linear equations. Then, the bias is derived from the 3-D angle of arrival measurements that take the measurement noise into account to improve the localization performance. Furthermore, a non-convex estimator is derived based on the Least Squares criterion. Finally, semi-definite relaxation and second-order cone relaxation are applied to transform the derived non-convex estimator into a convex one. We propose a semi-definite relaxation and second-order cone relaxation-based estimator which yields the best performance under a large measurement noise or a small measurement noise. The generalization of the proposed method for known transmit power can also be applied to the case when transmit power is not know. Theoretical analysis and computer simulations corroborate the superior performance of the proposed localization methods over the existing ones.

Cooperative RSS-Based Localization in Wireless Sensor Networks Using Relative Error Estimation and Semidefinite Programming

A new cooperative received signal strength-based localization algorithm is proposed which employs relative error estimation and semidefinite programming (SDP). First, the log-normal shadowing RSS measurement model is transformed into an equivalent multiplicative model. Then, a relative error estimation criterion is used with this model to develop a nonconvex estimator to approximate the maximum likelihood solution. Finally, semidefinite relaxation is applied to the nonconvex estimator to obtain an SDP estimator. The proposed algorithm is first derived for noncooperative RSS-based localization and then extended to the cooperative case. The Cramer–Rao lower bound is derived for cooperative RSS-based localization. Performance results are presented, which demonstrate that the proposed SDP estimator provides a significant improvement over existing localization methods.

Received signal strength–based target localization under spatially correlated shadowing via convex optimization relaxation

Received signal strength–based target localization methods normally employ radio propagation path loss model, in which the log-normal shadowing noise is generally assumed to follow a zero-mean Gaussian distribution and is uncorrelated. In this article, however, we represent the simplified additive noise by the spatially correlated log-normal shadowing noise. We propose a new convex localization estimator in wireless sensor networks by using received signal strength measurements under spatially correlated shadowing environment. First, we derive a new non-convex estimator based on weighted least squares criterion. Second, by using the equivalence of norm, the derived estimator can be reformulated as its equivalent form which has no logarithm in the objective function. Then, the new estimator is relaxed by applying efficient convex relaxation that is based on second-order cone programming and semi-definite programming technique. Finally, the convex optimization problem can be efficiently solved by a standard interior-point method, thus to obtain the globally optimal solution. Simulation results show that the proposed estimator solves the localization problem efficiently and is close to Cramer–Rao lower bound compared with the state-of-the-art approach under correlated shadowing environment.

RSS-Based Localization in WSNs Using Gaussian Mixture Model via Semidefinite Relaxation

Energy-based source localization methods are normally developed according to the channel path-loss models in which the noise is generally assumed to follow Gaussian distributions. In this letter, we represent the practical additive noise by the Gaussian mixture model, and develop a localization algorithm based on the received signal strength to achieve a maximum likelihood location estimator. By using Jensen’s inequality and semidefinite relaxation, the initially proposed nonlinear and nonconvex estimator is relaxed into a convex optimization problem, which can be efficiently solved to obtain the globally optimal solution. Besides, the corresponding Cramer–Rao lower bound is derived for performance comparison. Simulation and experimental results show a substantial performance gain achieved by our proposed localization algorithm in wireless sensor networks.

3-D Target Localization in Wireless Sensor Networks Using RSS and AoA Measurements

This paper addresses target localization problems in both noncooperative and cooperative 3-D wireless sensor networks (WSNs), for both cases of known and unknown sensor transmit power, i.e., $P_{T}$ . We employ a hybrid system that fuses distance and angle measurements, extracted from the received signal strength and angle-of-arrival information, respectively. Based on range and angle measurement models, we derive a novel nonconvex estimator based on the least squares criterion. The derived nonconvex estimator tightly approximates the maximum-likelihood estimator for small noise. We then show that the developed estimator can be transformed into a generalized trust region subproblem framework, by following the squared range approach, for noncooperative WSNs. For cooperative WSNs, we show that the estimator can be transformed into a convex problem by applying appropriate semidefinite programming relaxation techniques. Moreover, we show that the generalization of the proposed estimators for known $P_{T}$ is straightforward to the case where $P_{T}$ is not known. Our simulation results show that the new estimators have excellent performance and are robust to not knowing $P_{T}$ . The new estimators for noncooperative localization significantly outperform the existing estimators, and our estimators for cooperative localization show exceptional performance in all considered settings.

Distributed algorithm for target localization in wireless sensor networks using RSS and AoA measurements

This paper addresses target localization problem in a cooperative 3-D wireless sensor network (WSN). We employ a hybrid system that fuses distance and angle measurements, extracted from the received signal strength (RSS) and angle-of-arrival (AoA) information, respectively. Based on range measurement model and simple geometry, we derive a novel non-convex estimator based on the least squares (LS) criterion. The derived non-convex estimator tightly approximates the maximum likelihood (ML) one for small noise levels. We show that the developed non-convex estimator is suitable for distributed implementation, and that it can be transformed into a convex one by applying a second-order cone programming (SOCP) relaxation technique. We also show that the developed non-convex estimator can be transformed into a generalized trust region sub-problem (GTRS) framework, by following the squared range (SR) approach. The proposed SOCP algorithm for known transmit powers is then generalized to the case where the transmit powers are different and not known. Furthermore, we provide a detailed analysis of the computational complexity of the proposed algorithms. Our simulation results show that the new estimators have excellent performance in terms of the estimation accuracy and convergence, and they confirm the effectiveness of combining two radio measurements.

RSS-Based Localization in Wireless Sensor Networks Using Convex Relaxation: Noncooperative and Cooperative Schemes

In this paper, we propose new approaches based on convex optimization to address the received signal strength (RSS)-based noncooperative and cooperative localization problems in wireless sensor networks (WSNs). By using an array of passive anchor nodes, we collect the noisy RSS measurements from radiating source nodes in WSNs, which we use to estimate the source positions. We derive the maximum likelihood (ML) estimator, since the ML-based solutions have particular importance due to their asymptotically optimal performance. However, the ML estimator requires the minimization of a nonconvex objective function that may have multiple local optima, thus making the search for the globally optimal solution hard. To overcome this difficulty, we derive a new nonconvex estimator, which tightly approximates the ML estimator for small noise. Then, the new estimator is relaxed by applying efficient convex relaxations that are based on second-order cone programming and semidefinite programming in the case of noncooperative and cooperative localization, respectively, for both cases of known and unknown source transmit power. We also show that our approaches work well in the case when the source transmit power and the path loss exponent are simultaneously unknown at the anchor nodes. Moreover, we show that the generalization of the new approaches for the localization problem in indoor environments is straightforward. Simulation results show that the proposed approaches significantly improve the localization accuracy, reducing the estimation error between 15% and 20% on average, compared with the existing approaches.

On the MSE Properties of Empirical Bayes Methods for Sparse Estimation

Popular convex approaches for sparse estimation such as Lasso and Multiple Kernel Learning (MKL) can be derived in a Bayesian setting, starting from a particular stochastic model. In problems where groups of variables have to be estimated, we show that the same probabilistic model, under a suitable marginalization, leads to a different non-convex estimator where hyperparameters are optimized. Theoretical arguments, independent of the correctness of the priors entering the sparse model, are included to clarify the advantages of our non-convex technique in comparison with MKL and the group version of Lasso under assumption of orthogonal regressors.

Received Signal Strength-Based Wireless Localization via Semidefinite Programming: Noncooperative and Cooperative Schemes

The received signal strength (RSS)-based approach to wireless localization offers the advantage of low cost and easy implementability. To circumvent the nonconvexity of the conventional maximum likelihood (ML) estimator, in this paper, we propose convex estimators specifically for the RSS-based localization problems. Both noncooperative and cooperative schemes are considered. We start with the noncooperative RSS-based localization problem and derive a nonconvex estimator that approximates the ML estimator but has no logarithm in the residual. Next, we apply the semidefinite relaxation technique to the derived nonconvex estimator and develop a convex estimator. To further improve the estimation performance, we append the ML estimator to the convex estimator with the result by the convex estimator as the initial point. We then extend these techniques to the cooperative localization problem. The corresponding Cramer-Rao lower bounds (CRLB) are derived as performance benchmarks. Our proposed convex estimators comply well with the RSS measurement model, and simulation results clearly demonstrate their superior performance for RSS-based wireless localization.

Shape-restricted inference for Lorenz curves using duality theory

We propose a new methodology for estimating curves under a convexity restriction based on Fenchel duality and wavelet approximations. In contrast to approaches where a possibly non-convex estimator is convexified at a second stage, our procedure allows us to construct directly an estimator with a convex shape. The method is applied to the estimation of the Lorenz curve. Applications to estimation of average value at risk, as well as multivariate generalisations to Lorenz surfaces are mentioned. We show asymptotic efficiency which demonstrates that the convexity is achieved at no extra cost.