On The Grouping Effect Of The L₁-l₂ Models

With the rapid development of information technology,in order to obtain more effective information from the collected data,data processing has become an urgent problem in many fields of modern science and engineering.Due to the diversity of data sources,the data in computational biology,medical imaging,natural language processing and other fields have the characteristics of high dimensional data.Valuable features of high dimensional data are usually hidden in different low dimensional subspaces of the original feature space.Feature selection is the process of selecting related feature subset from original feature set,which plays an important role in high dimensional data processing.In recent years,feature selection methods based on sparse learning have been widely studied,and different regularization methods have been proposed.Due to the high efficiency of compressed sensing theory in signal processing,the cost of data storage and transmission is significantly reduced.As a regularized least square regression method,Elastic Network combines the characteristics of 1-norm and 2-norm,and is widely used in learning and variable selection.In linear regression system,the coefficients corresponding to highly correlated predictors have little difference.When the coefficients have the same sign,this small difference can be estimated quantitatively in theory.Even if the coefficients have different signs,the same estimation is correct in the Elastic Network.When the l₁+l₂ model fits the data well,the empirical approximation error can be used to improve the estimation.In addition,we find that the l₁-l₂ model also has grouping effect when the coefficient sign is the same.The main work of this paper includes the following aspects.Firstly,the condition of l₁+l₂ minimization model satisfying grouping effect is analyzed and the corresponding theorem is given.For the constrained l₁+l₂ minimization model,it is proved that the grouping effect holds when the entries of vectors have different symbols.See theorem 3.3 and theorem 3.4 for details.Then we study the mathematical properties of the l₁-l₂ model with highly correlated column measurement matrix.It is theoretically proved that the l₁-l₂ model satisfies the grouping effect.Finally,the stability analysis of l₁-l₂ model based on sparse approximation property is given by using high correlation Gaussian random matrix.