# Indoor Localization Using Semi-Supervised Manifold Alignment with Dimension Expansion
## 2 Related Work  
As we know, the RSS fingerprinting-based indoor localization requires the site survey of the target environment, which is always time-consuming and labor-intensive. To solve this problem, the semi-supervised learning approach has attracted significant attention by using a small number of calibrated locations to achieve the satisfactory localization performance [24]. As one of the most popular representatives in semi-supervised learning, the manifold alignment approach is much preferred for indoor localization. This approach is based on the assumption that the measurements collected at nearby locations, which are used for localization, like the RSS, TOA and AOA, are featured with good neighborhood relations in a low-dimensional manifold [25]. By perturbing the local geometries of physical coordinates, the manifold alignment approach in [26] locates the target with a limited number of calibrated locations. The authors in [27] give a solution to the fast radio map construction in indoor WLAN environment by using the crowdsourced manifold alignment approach. Additionally, the locations of the target are estimated by matching the newly-collected RSS measurements against the radio map in a low-dimensional manifold. By employing the unsupervised manifold alignment approach, the authors in [28] utilize the WLAN RSS fingerprints and newly-collected RSS measurements to construct a source dataset and meanwhile rely on the coordinates of RSS fingerprints to construct a destination dataset. After performing the transformation from the source into destination datasets, the target is located based on the neighbor matching of the transformed data with respect to the newly-collected RSS measurements and coordinates of RSS fingerprints in the common manifold. The authors in [29] apply the local tangent space alignment and Multidimensional Scaling (MDS) approaches to develop a new manifold learning approach, by which the pairwise distance between every two RSS, TOA and TDOA measurements is considered to construct various local maps, and the overlap of different local maps is aligned to obtain the relative global coordinates in physical space. The manifold flattering approach for anchor-less localization is proposed in [30]. This approach relies on the Euclidean distance matrix to gradually infer the unknown distances among the newly-collected RSS measurements and then applies the MDS approach to calculate the relative locations of the target. The authors in [31] propose to use the semi-supervised Laplacian regularized least squares approach for localization. Specifically, this approach depends on an alignment criterion to learn the appropriate kernel function with respect to the similarities between different anchors and then uses the manifold regularization-based semi-supervised learning approach to locate the target. A motion sensor-free localization approach is discussed in [32], in which the floor plan and temporal RSS sequences are used to perform the graph matching-based manifold alignment. The authors in [33] embed the RSS measurements collected on different floors into a common low-dimensional manifold and then align the labeled and unlabeled RSS measurements for localization. In all, the previous literature mainly focuses on the similarity of the calibrated fingerprints in the low-dimensional space and involves the online RSS measurement in the objective function, which sometimes cannot preserve the neighborhood relations of the calibrated fingerprints well and guarantee the stability of transformation relations with respect to the physical coordinates and the related RSS measurements. Different from the approaches mentioned above, the proposed semi-supervised manifold alignment with dimension expansion is based on an innovative objective function that can better preserve the neighborhood relations of the augmented physical coordinates and the corresponding RSS measurements at nearby locations.