In order to reduce complexity of construction of indoor positioning fingerprint database and improve the positioning accuracy, an indoor fingerprint positioning algorithm based on matrix completion under the 5G ultra-dense network is proposed. In the offline database construction stage,the algorithm first uses the K-nearest Neighbor(KNN) interpolation method to complete the matrix of part of the fingerprint database to construct a complete database. Secondly,the sparse auto-encoder is used to extract the sparse features of the fingerprint database, and the high-dimensional received signal strength indication (RSSI) signal is reduced. In the online fingerprint matching stage,the weighted KNN algorithm is used to estimate the coordinates of the point to be located. After experimental simulation,the average relative error of the algorithm to reconstruct the fingerprint database is 0.31% . Compared with that of the traditional KNN fingerprint matching algorithm,the average error is reduced by 24.41% .