Optimal Low-Rank QR Decomposition with an Application on RP-TSOD

Low-rank matrix approximation has many applications, e.g., denoising, recommender systems and image reconstruction. Recently, a Randomized Pivoted Two-Sided Orthogonal Decomposition (RP-TSOD) was developed to exploit the randomization in approximating a high-dimensional matrix using QR decomposition. Instead of random projection, we propose to optimize the projection matrix for low-rank QR decomposition with the target of minimizing the approximation error. A method based on gradient descent is developed to derive optimal projections. The developed techniques can be used in not only RP-TSOD, but also other decompositions. Experimental results on both synthetic data and real data show that the proposed method could more accurately approximate a high-dimensional matrix than RP-TSOD.