## 1 Introduction  
Missing data is a problematic issue in many applications. This problem may arise from incomplete data collection, unavailability of data, or corrupted data during the transmission process. Recovering missing data accurately is essential, especially for the purposes of data analysis and prediction. Various approaches have been proposed for reconstructing missing data. In [1], the deep convolution neural network was proposed to reconstruct missing data for a remote sensing image. A deep learning approach was used for irregularly and regularly missing data reconstruction in [2]. The concept of fuzzy similarity was used for recovering data in multidimensional time series in [3]. The long short-term memory(LSTM) approach was introduced in [4] for reconstructing time series Landsat images. A methodology based on principles of matrix theory and pseudo-inverses was proposed in [5] for data reconstruction in erasure codes. In [6], a method of tensor completion based on nuclear norm minimization was introduced for 5D seismic data reconstruction. Other approaches for approximating missing data include the regression method [7], K-nearest neighbors (KNN) [8, 9], principal component analysis (PCA) [10], and least-squares approximation [11, 12, 13]. These approaches are mostly specific to certain problems. The goal of this paper is to improve a general framework based on a least-squares method for estimating missing data with an optimal low-rank basis in je Euclidean norm, which is also called proper orthogonal decomposition (POD) basis. POD is also known as, for example, Karhunen-Loeve decomposition (KLD), principal component analysis (PCA), or singular value decomposition(SVD). POD has been successfully used in many applications, e.g., [14, 15, 16, 17, 18], since it can provide an approximation from the basis that extracts the dominant characteristic of the existing data. POD has been used in many applications, such as aerodynamic flow fields [19], chemical engineering [20, 21],mechanical engineering [22; 23], image processing [24], and optimization of water flooding reservoir [25]. In general, the POD basis can be computed by using singular value decomposition (SVD). However, for high-dimensional data, obtaining SVD by a traditional approach may be computationally costly and memory intensive. Randomized SVD (rSVD) [26] is considered in this work to reduce this computational complexity. The rSVD algorithm, introduced by [PERSON] et al. [26], employs randomness to obtain a smaller matrix from a high-dimensional data matrix. The smaller matrix is then used to compute the low-dimensional basis. The rSVD algorithm is shown to be computationally efficient for approximating matrices with low-rank structures. The algorithm is divided into two stages. In the first stage, random sampling is used to obtain a reduced matrix whose range approximates the range of the original matrix. A small matrix can be obtained from the orthogonal projection onto the basis of the reduced matrix from the first stage. In the second stage, the smaller matrix is then inexpensively factorized. Randomized methods have been used in many applications, such as low-rank SVD of a large matrix [27], image compression [28; 29], and image reconstruction [30]. The implementation and extension of the rSVD have been investigated in many works, e.g., [31; 32; 33; 34]. This work applies rSVD to reconstruct missing data components with the notion of least-squares approximation. The least-squares method has been used in missing data reconstruction in previous work, e.g., [11; 12; 13], which is mainly based on employing all remaining available data. In the case of high-dimensional data, the reconstruction process might require high-complexity computation. To accelerate this approximation process further, this work also employs certain important and relevant available data components, which are selected through a greedy iterative procedure called the discrete empirical interpolation method (DEIM).