Stable Generalized Low Rank Approximations Of Matrices

As a commonly-used processing technique for 2-dimensional datasets,Generalized Low Rank Approximations of Matrices?GLRAM?can be employed to reduce dimensionality,remove noise and complete the missing entries.In the existing work,the input image is usually reshaped into a vector before the low rank decomposition.It has been confirmed that this process will destroy the inherent2-dimensional correlation in the image.Therefore,this thesis proposes a new low-rank approximation algorithm from another perspective.This thesis studies GLRAM and robust GLRAM.In order to enhance the robustness and stability of GLRAM,Stable Generalized Low Rank Approximation of Matrices?SGLARM?is proposed.The proposed model decomposes each data matrix into the sum of low-rank components,sparse noise,and dense Gaussian noise.Meanwhile,the missing entries are also considered.To recover the low rank matrix,an optimization problem minimizing matrix ₁l-norm and Frobenious norm is established.For the established minimization problem,the Alternating Direction Method of Multipliers?ADMM?is designed to solve it.The experimental results on synthetic data sets and image data sets verify the feasibility and effectiveness of the proposed method.Considering the sparse representation of the low rank components,this thesis presents a new improved algorithm,i.e.,Majorizing Stable Generalized Low Rank Approximations of Matrices?MSGLRAM?.Compared with SGLRAM,the new algorithm imposes sparsity on all low rank matrices.This optimization algorithm is applied to artificial data sets and face image data sets.The experimental results show that the model is feasible.