A Study Of Huber Matrix Regression:Theory And Algorithms

With the advent of the big data era,we are facing with more and more complex data.Data in the form of matrix is emerging from scientific research and practical application fields,such as gene expression analysis,brain neural network,finance,e-conomics,machine learning and artificial intelligence,medical imaging diagnosis and treatment of diseases,risk management and so on.At present,most of the processing of these data is based on the assumption of homoscedasticity and then the least squares model is used for statistical analysis.In fact,a lot of them are not homoscedastic.At this time,the least squares model cannot explain these data well.In this case,it is nat-ural to consider robust methods such as quantile regression.In practice,however,it is not clear whether the data is heavily tailed or contains outliers.At this point,it is a good idea to consider using the Huber function as a loss function.Huber function is a combination of the quadratic function and the absolute value function,which is the smoothing function of the absolute value function.In the optimization,smoothing is more conducive the design of optimization algorithm.In addition,there are structural features in the data,such as sparsity of elements,sparsity of prediction variables,low-rank,multicollinearity and so on.In this thesis,the Huber matrix regression model is established through regularization technology.Then,we make a study on the statisti-cal property and design optimization algorithm for these models.At last,we use these models to analyze synthetic and real data.For low-rank and heavy-tailed matrix data,we establish the nuclear norm regular-ized Huber matrix regression model.By using the decomposability of nuclear norm,local restrict strongly convexity and nearly low-rank of Huber loss function,we give the upper bound of the risk bound.Then an accelerated proximal gradient algorithm(mAPG)with iteration complexity of O((?))is designed to estimate the coeffi-cient matrix of this model.Finally,we use this model to analyze simulated data and Norwegian paper quality data.The numerical results show that the nuclear norm regu-larized Huber matrix regression model can give better results.For low-rank and heavy-tailed matrix data with multicollinearity,we establish the low-rank elastic-net regularized Huber matrix regression model and prove that it pos-sesses the grouping property.Based on some mild conditions on noise and design matrix,an upper risk bound of the estimator is obtained.Based on the mAPG algorithm proposed in Chapter 2,we consider continuation technique and truncation technique to accelerate mAPG algorithm.By analyzing simulated data and Arabidopsis thaliana data,we found that the low-rank elastic-net regularized Huber matrix regression model can deal with collinearity in heavy-tailed data well.For heavy-tailed matrix data with few material predictors and multicollinearity,we established the row-sparse and elastic-net regularized Huber matrix regression model and proved its group property theoretically.The risk upper bound is established under some assumptions.In addition,an accelerated proximal sub-gradient algorithm with iteration complexity of O(?)is designed to solve this model.Through sim-ulations and analysis of polyethylene data,we found that the row-sparse and elastic-net regularized Huber matrix regression model could not only deal with collinearity in heavy-tailed data,but also select material predictors.In short,for the matrix regression problem with different structures,we proposed the regularized Huber matrix regression models,studied the statistical properties,de-signed effective optimization algorithms and gave the convergence analysis,and veri-fied the effectiveness and theoretical properties of the model through numerical experi-ments.