# Indoor Localization Using Semi-Supervised Manifold Alignment with Dimension Expansion  
[PERSON] 1,2, [PERSON] 3,4, [PERSON] 3 and [PERSON] 3  
Chongqing Key Lab of Mobile Communications Technology, Chongqing University of Posts and Telecommunications, Chongqing 400065, China; [EMAIL_ADDRESS] (M.Z. ); [EMAIL_ADDRESS] (Z.T. ); [EMAIL_ADDRESS] (Y.W.) Correspondence: [EMAIL_ADDRESS]; Tel. : +86-23-6246-0295 Current address: No. 2 Chongwen Road, Nan'an District, Chongqing 400000, China. These authors contributed equally to this work. Footnote 1: [[https://github.com](https://github.com)]([https://github.com](https://github.com))  
The measurements, like Time Of Arrival (TOA) [16], Time Difference Of Arrival (TDOA) [17] and Angle Of Arrival (AOA) [18], as well as the propagation model [19] and Received Signal Strength (RSS) fingerprinting [20], have attracted significant attention for WLAN localization. As the most frequently-used measurement, RSS fingerprinting [21] is featured with no extra hardware requirement. However, the RSS fingerprinting involves time-consuming and labor-intensive location calibration, which significantly limits its application to the large-scale indoor environment. To solve this problem, the semi-supervised learning approach [22] has been studied intensively for indoor localization based on a limited number of calibrated locations with known physical coordinates and the corresponding collected RSS measurements. On the one hand, the semi-supervised learning approach requires a small number of calibrated locations, which can effectively relieve the load for the off-line site survey of the target environment. On the other hand, the satisfactory localization performance is normally reached by the semi-supervised learning approach without any motion sensor assistance. As one of the most popular representatives in semi-supervised learning, the manifold alignment approach has been carefully investigated [23]. The conventional manifold alignment approaches generally assume that the measurements collected at nearby locations, which are used for localization, are featured with good neighborhood relations in a low-dimensional manifold. Based on this, the solution to the objective function in manifold alignment is calculated as the closed-form in the low-dimensional manifold. Then, the target locations are estimated by the closed-form solution to the objective function in the low-dimensional manifold. This approach heavily depends on the similarity of the calibrated fingerprints in the low-dimensional space, which cannot well preserve the neighborhood relations of the calibrated fingerprints, and meanwhile, its objective function involves the online RSS measurements, which decrease the stability of transformation relations among the calibrated fingerprints. Different from the conventional manifold alignment approaches, we propose a new one that can better preserve the neighborhood relations of the physical coordinates and the corresponding RSS measurements at nearby locations. The basic idea of the proposed approach is to perform the augmentation of physical coordinates to make the dimensions of the augmented physical coordinates and RSS measurements equal and then construct the two transformation matrices with respect to the augmented physical coordinates and the corresponding raw RSS measurements. After that, the target locations can be estimated by using the neighbors in the common manifold transformed from the augmented physical coordinates, as well as the newly-collected RSS measurements. In sum, the three main contributions of this paper are summarized as follows. First of all, an innovative objective function in manifold alignment is constructed based on the augmented physical coordinates and the corresponding RSS measurements, which can be easily generalized to the scenario with multiple types of measurements used for localization.