针对儿童孤独症的临床数据填补，提出了一种基于交替方向乘子(ADMM)算法的高秩矩阵填充(HRMC)算法.该算法考虑到数据中的不同参数具有不同的重要性，在算法设计时将重要参数与非重要参数赋予不同权重.在案例分析中，利用生成的测试数据寻找最优的参数配置，并且将该算法应用到实际孤独症数据的填充中, 填充结果与参数化填充方法和低秩矩阵填充方法的结果进行对比.结果显示：所提出的算法在矩阵填充精度方面显著优于其他算法，并可应用于实际的数据清洗与处理过程中.
To solve the clinical data recover problem of autism spectrum disorders (ASDs), a high-rank matrix completion (HRMC) algorithm based on alternating direction method of multipliers (ADMM) was proposed. Under consideration of different parameters with different significance, the important parameters and unimportant ones were attached with various weights. In a case study, test data were generated to find the optimal parameters. Furthermore, the algorithm was applied on practical ASD clinical data. The results show that the algorithm performs better in comparison with other parameterized algorithms and normal matrix completion algorithm, which indicates that it can be applied in practical data cleaning and processing.
［1］LAI M C, LOMBARDO M V, BARON-COHEN S. Autism［J］. Lancet, 2014, 383(9920): 896-910.
［2］CANDS E J, RECHT B. Exact matrix completion via convex optimization ［J］. Foundations of Computational Mathematics, 2009, 9(6): 717-772.
［3］XU Y Y, YIN W T, WEN Z W, et al. An alternating direction algorithm for matrix completion with nonnegative factors［J］. Frontiers of Mathematics in China, 2012, 7(2): 365-384.
［4］RECHT B, R C. Parallel stochastic gradient algorithms for large-scale matrix completion［J］. Mathematical Programming Computation, 2013, 5(2): 201-226.
［5］曾广翔. 面向推荐系统的矩阵填充算法研究［D］. 合肥: 中国科学技术大学, 2015.
ZENG Guangxiang. Research of matrix completion algorithm for recommendation system［D］. Hefei: University of Science and Technology of China, 2015.
［6］YEGANLI S F, YU R. Image inpainting via singular value thresholding［C］//Signal Processing and Communications Applications Conference (SIU). Haspolat: IEEE, 2013: 1-4.
［7］LI W, ZHAO L, LIN Z J, et al. Non-local image inpainting using low-rank matrix completion［J］. Computer Graphics Forum, 2014, 34(6): 121-122.
［8］JI H, LIU C, SHEN Z, et al. Robust video denoising using low rank matrix completion［C］//IEEE Computer Society Conference on Computer Vision and Pattern Recognition. San Francisco: IEEE, 2012: 1791-1798.
［9］ROOZGARD A, BARZIGAR N, VERMA P, et al. 3D medical image denoising using 3D block matching and low-rank matrix completion［C］//2013 Asilomar Conference on Signals, Systems and Computers. CA, USA: IEEE, 2013: 253-257.
［10］ELHAMIFAR E. High-rank matrix completion and clustering under self-expressive models［C］//Advances in Neural Information Processing Systems. Barcelona: Curran Associates, Inc. 2016: 73-81.
［11］FAN J C, CHOW T W S. Sparse subspace clustering for data with missing entries and high-rank matrix completion ［J］. Neural Networks, 2017, 93: 36-44.
［12］BOYD S. Distributed optimization and statistical learning via the alternating direction method of multipliers ［J］. Foundations and Trends in Machine Learning, 2010, 3(1): 1-122.
［13］BECK A, TEBOULLE M. A fast iterative shrinkage-thresholding algorithm for linear inverse problems ［J］. SIAM Journal on Imaging Sciences, 2009, 2(1): 183-202.
［14］CAI J F, CANDS E J, SHEN Z W, et al. A singular value thresholding algorithm for matrix completion ［J］. SIAM Journal on Optimization, 2010, 20(4): 1956-1982.
［15］YANG C Y, ROBINSON D, VIDAL R. Sparse subspace clustering with missing entries ［J］. Proceedings of the 32nd International Conference on Machine Learning, PMLR, 2015, 37: 2463-2472.
