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A common approach for compressing large-scale data is through matrix sketching. In this work, we consider the problem of recovering low-rank matrices from two noisy linear sketches using the double sketching scheme discussed in Fazel et al. [Compressed sensing and robust recovery of low rank matrices, in Proceedings of the 42nd IEEE Asilomar Conference on Signals, Systems and Computers, 2008, pp. 1043-1047], which is based on an approach by Woolfe et al. [Appl. Comput. Harmon. Anal., 25 (2008), pp. 335-366]. Using tools from nonasymptotic random matrix theory, we provide the first theoretical guarantees characterizing the error between the output of the double sketch algorithm and the ground truth low-rank matrix. We apply our result to the problems of low-rank matrix approximation and low-tubal-rank tensor recovery. [ABSTRACT FROM AUTHOR]

Video synthetic aperture radar (SAR) is attracting more and more attention because of its continuous imaging capability for ground scene of interest under any weather conditions and any time of the day. To reduce the sampling amount of video SAR, image processing can be formulated into a low-tubal-rank tensor recovery problem. In this article, we proposed an orthogonal rank-1 tensor pursuit (ORTP) algorithm to solve the low-tubal-rank tensor recovery problem in video SAR imaging. The proposed ORTP algorithm is an extension of the orthogonal rank-1 matrix pursuit algorithm in the matrix sensing problem from the matrix case to the tensor case under a tubal-rank model. It is capable of reconstructing the target tensor efficiently without requiring any prior information about the prespecified or pre-estimated tensor tubal-rank value. To achieve this, rank-1 basis tensors and weight tensors of the target tensor are estimated iteratively, and the residual error between the observed tensor and the estimated tensor through linear mapping is utilized as the stop condition. We theoretically prove the convergence and correctness of the proposed ORTP method. The methodology was tested on synthetic data, real video data, and video SAR data. These tests show that the proposed approach outperforms other video SAR imaging algorithms and low-rank tensor recovery algorithms.