# Beyond Stochastic Gradient Descent for Matrix Completion Based Indoor Localization
### Related Works  
As mentioned in the previous section, it is difficult, from available RSSI measurements, to acquire all the pairwise distances between sensors and to obtain a full connected squared distance matrix so called the Euclidean distance matrix (EDM) [18]. This is due to the limitation of communication range and multipath effects. To provide a reliable distance information to each sensor node, the matrix completion is then proposed and its role is to recover the complete Euclidean distance matrix EDM from an incomplete matrix. Several algorithms and approaches have been proposed and studied to approximate the missing distances for RSSI measurements based systems. Authors in [14] formulate the matrix completion problem using the least squared minimization. To solve the minimization problem, they introduce a modified iterative Newton's algorithm [19] to optimize the objective function. This solution can be very fast if the parameters are well chosen. The main drawback is that the result is very sensitive to initialization. It can be considered as a good initial location estimation for other fine localization algorithms. Other existing approaches based on Singular Value Thresholding (SVT) [20], Accelerated Proximal Gradient (APG) [21] and Augmented Lagrange Multiplier (ALM) [22] have been proposed in recent years. SVT method which relies on the Singular Value Decomposition (SVD) of a low rank matrix is associated to poor completion performance and slow convergence rate. Motivated by the above reasons, ALM and APG methods have been proposed. However, these methods depend on the choice of relevant parameters and are sensitive to noise. To overcome this problem, a regularized matrix completion model is proposed in [23] introducing the multivariate function Bregman divergence to solve the EDM problem. The major drawback of such a centralized method is its computational cost. The aforementioned methods are all based on RSSI. As an alternative to RSSI measurements, [PERSON] et al. focus on connectivity/distance information setting a routing protocol in place [15; 16]. They formulate the matrix completion problem as an unconstrained optimization in Smooth Riemannian manifold. Then, the nonlinear conjugate gradient algorithm is applied on the smooth Riemannian manifold. This approach recovers the EDM in noiseless and noisy environments. However, it focused only on two dimensional position information. Authors in [24] cover the unknown measurements under the existence of noise which is classified into three categories. This approach achieves a good localization accuracy compared to other existing ones. However, it is a centralized localization approach. Motivated by the above reasons, based on RSSI measurements, we solve the minimization problem using advanced algorithms to reduce the running time and the computational complexity while ensuring a good localization accuracy. After completing the EDM, the second step is to estimate the location of sensors. For this, the completed pairwise distance information can be transformed into the estimated coordinates of sensor nodes by applying a factorization process [14]. This factorization requires that the matrix is Semi Definite Positive (SDP). The SDP problem can be efficiently and accurately solved by CVX toolbox on Matlab [25]. However, if the matrix is not SDP, the problem can also be solved by introducing Semidefinite Relaxations (SDR) [26; 27]. The SDR based localization scheme in [26] relaxes the non convex localization into a convex one and marks a good localization accuracy. However,the computational complexity of SDR-based techniques is closely related to the problem size. Therefore, this algorithm is adapted for medium size network only due to the highly required running time. To reduce the running time, a weighted semidefinite relaxation localization method is carried out in [27]. It aims to improve the accuracy of localization, but it is still not suitable for dealing with large networks.