低秩张量恢复及应用

As the latest development of compressed sensing theory, Low-Rank Matrix Recovery (LRMR) employs rank to measure the complexity of matrices and it usually recovers the low-rank components and sparse noises by taking the sparsity into account. Nowadays, LRMR has been widely applied in the fields of computer vision, pattern classification, machine learning, recommendation system, system identification and so on. Considering that multi-linear signals are ubiquitous, the project plans to research the theory, algorithms and applications of Low-Rank Tensor Recovery (LRTR). The main research contents of the project are listed as follows. Firstly, we establish the multi-linear robust principal component analysis model, construct a fast and effective algorithm to solve it and discuss the convergence performance. Secondly, for the general LRTR, we propose a tensor nuclear norm minimization model, present a solving strategy which combines the low-rank tensor approximations and augmented Lagrange multipliers, and then the conditions are given for exactly recovering the low-rank components. Finally, we provide proper probability distribution assumptions on low-rank components, noise, the matrix singular values and representation coefficients, build a LRTR model based on maximum a posteriori probability, and solve the model by utilizing the expectation maximization or augmented Lagrange multipliers. The project will overcome the defect of traditional vectorization representation of tensor signals, and discuss the relationship between LRTR and compressed sensing. It has significant theoretical contribution in sampling, denoising, and dimensionality reduction of tensor signals. Moreover, it also has important application value in computer vision, pattern classification, etc..

作为压缩感知理论的最新进展，低秩矩阵恢复使用秩来度量复杂度，并常结合稀疏性来恢复低秩成分和稀疏噪声，它已被广泛地应用在计算机视觉、模式分类、机器学习、推荐系统和系统辨识等领域中。鉴于多线性信号是普遍存在的，本项目欲研究低秩张量恢复的理论、算法及应用，主要内容如下：建立多线性鲁棒主成分分析模型，构造求解此模型的快速有效的算法，分析算法的收敛性；对于一般形式的低秩张量恢复，提出张量核范数最小化模型，结合低秩张量逼近和增广拉格朗日乘子法来求解它，并给出低秩成分能被恢复的条件；对低秩成分、噪声、矩阵奇异值和表示系数等做适当的概率假设，基于最大后验概率建立低秩恢复模型，并使用期望最大化或增广拉格朗日乘子法求解所建模型。该项目拟克服传统的张量信号向量化的不足，探讨低秩张量恢复与压缩感知的关系，在多线性信号采样、去噪和维数约简等方面具有重要的理论意义，在计算机视觉和模式分类等领域中具有重要的应用价值。

低秩张量恢复是机器学习和计算机视觉领域中处理多线性数据的一类重要的技术，其目的是根据数据集的近似低秩性质来恢复低秩成分、补全丢失元素、分离稀疏噪声。在本项目支持下，我们对低秩张量恢复的模型和算法进行了系统的研究，取得了一系列成果，也为进一步研究概率低秩张量恢复奠定了基础。取得的主要成果如下。. （1）对于广义低秩张量恢复模型，设计了交替方向乘子算法，给出了算法的弱收敛性。（2）针对张量补全问题，提出了基于Tucker分解的最小二乘算法。与张量核范数最小化算法相比，所提方法使用矩阵QR分解，避免了奇异值分解，从而降低了计算复杂度。（3）建立了鲁棒广义低秩矩阵逼近模型，提出了相应的求解算法，证明了算法的收敛性，并将模型推广到张量情形。（4）对于含有丢失元素的数据，提出了基于稀疏表示的分类方法，该方法对大量的丢失元素具有鲁棒性。（5）考虑到视频序列在时间上的连续性与不完整性，提出了正则化的鲁棒主成分分析，该方法能更有效地恢复出前景。（6）对主要的低秩矩阵分解的概率模型进行了归纳与总结，并探讨了概率张量分解模型。. 该项目共发表相关论文23篇，其中SCIE收录12篇，EI收录5篇。培养硕士研究生2人，研究成果在机器学习和计算机视觉等领域具有一定的理论意义和应用价值。