Matrix (chemical analysis)

Distance matrix

Algorithm

Combinatorics

Euclidean distance

Physics

Mathematical optimization

Engineering

Euclidean geometry

Mathematical analysis

Artificial intelligence

Tensor (intrinsic definition)

Composite material

Economics

Gaussian

Rank (graph theory)

Euclidean distance matrix

Materials science

Geometry

Convergence (economics)

Matrix completion

Riemannian manifold

Mathematics

Computer science

Economic growth

Manifold (fluid mechanics)

Quantum mechanics

Mechanical engineering

Conjugate gradient method

Low-rank approximation

IEEE Transactions on Communications

Location awareness, providing the ability to identify the location of sensor, machine, vehicle, and wearable device, is a rapidly growing trend of hyper-connected society and one of the key ingredients for the Internet of Things (IoT) era. In order to make a proper reaction to the collected information from things, location information of things should be available at the data center. One challenge for the IoT networks is to identify the location map of whole nodes from partially observed distance information. The aim of this paper is to present an algorithm to recover the Euclidean distance matrix (and eventually the location map) from partially observed distance information. By casting the low-rank matrix completion problem into the unconstrained minimization problem in a Riemannian manifold in which a notion of differentiability can be defined, we solve the low-rank matrix completion problem using a modified conjugate gradient algorithm. From the convergence analysis, we show that localization in Riemannian manifold using conjugate gradient (LRM-CG) converges linearly to the original Euclidean distance matrix under the extended Wolfe's conditions. From the numerical experiments, we demonstrate that the proposed method, called LRM-CG, is effective in recovering the Euclidean distance matrix.

Localization of IoT Networks via Low-Rank Matrix Completion