Sparse And Low Rank Tensor Optimization Methods And Applications

As a high-order and high-dimensional data structure,tensor has been widely used in video surveillance,data mining,computer vision,signal processing,engineering and statistics.In recent years,due to the continuous expansion of the data scale generated by practical problems,in which sparsity and low-rank are important characteristics,the theoretical algorithm and application research of sparse low-rank tensor optimization based on tensor decomposition have become very challenging frontier topics in the optimization field.In this paper,the sparse and low-rank tensor optimization method is applied to video background subtraction and internet traffic anomaly detection.To address background subtraction from compressive measurements(BSCM),we firstly propose a new Tucker decomposition-based sparse tensor optimization method,which makes full use of the spatio-temporal features embedded in the video.The ₀-norm in the objective function is used to constrain the sparseness of the spatio-temporal structure of video foreground,which enhances the spatio-temporal continuity and im-proves the accuracy of foreground detection.The orthogonality constraints on factor matrices in low-rank Tucker decomposition are used to characterize the spatio-temporal correlation of video background,which enhances low-rank characterization and makes better background estimation(see Section 2.1).Secondly,this paper analyzes the model in the theory,then establishes the optimality conditions for the proposed sparse tensor optimization problem(see Section 2.2)and a hard-thresholding based alternating di-rection method of multipliers(HT-ADMM)is designed for the proposed model(see Section 2.3).Finally,comprehensive experiments are conducted on real-world video datasets to demonstrate the superiority of our model and the effectiveness of the algo-rithm(see Section 2.4).For the internet traffic anomaly detection(ITAD)problem,we firstly formulate it by a sparse and low-rank tensor optimization model,taking into full consideration the intrinsic and potential properties including the sparsity of anomalies,the low-rankness and temporal stability and periodicity of the normal traffic data(see Section 3.1).Sec-ondly,this paper analyzes the model.Although the resulting optimization model is non-convex and discontinuous due to the involved ₀-norm and the tensor rank func-tion,optimality analysis via stationarity is established(see Section 3.2),based on which an efficient proximal gradient method with theoretical convergence to stationary points is designed(see Section 3.3).Finally,numerical experiments on internet traffic trace data Abilene and G`EANT demonstrate the high efficiency of our proposed sparse and low-rank tensor-based approach(SLRTA)for ITAD(see Section 3.4).