Robust Bilinear Probabilistic Principal Component Analysis

Probabilistic principal component analysis(PPCA)is a probabilistic latent variable model based on normal distribution,which is a widely used technique for dimensionality reduction on 1-D data.To apply PPCA to 2-D data where the observations are matrices,we must first vectorize the data then apply PPCA to the 1-D data.However,the size of the vector data after straightening will be very high,PPCA will suffer from the curse of dimensionality.Therefore,the bilinear probabilistic principal component analysis(BPPCA)is proposed,BPPCA is a bilateral extension of the PPCA and uses separated covariance.This model directly reduces the dimension on the original matrix data structure,overcoming the problem of the curse of dimensionality.However,both PPCA and BPCCA are probability models based on normal distribution,and the model under the assumption of normal distribution is more sensitive to outliers.When the data contains outliers,PPCA and BPCCA,which are models based on the normal distribution assumption,may have large estimation bias,and the effect of reducing the dimension is not ideal.While the student t distribution is similar to the normal distribution,the tail of the student t distribution is heavier than the normal distribution,and the student t distribution contains degrees of freedom,therefore,the modeling of the t distribution has been proven to have more excellent performance in practical applications.Therefore,based on the matrix-variate t distribution,this paper proposes a new model called robust bilinear probabilistic principal component analysis(t BPPCA),and four maximum likelihood estimation algorithms EVME,ECM,AECM1 and AECM2 are proposed to fit the model to be robust existing model to broaden its applicability.The ECME algorithm is similar to the ECM algorithm,the difference is only that ECME updates the degrees of freedom by maximizing the observed data likelihood function,while ECM updates the degrees of freedom by maximizing the expected complete data likelihood function.The AECME1 algorithm is also similar to the AECM2 algorithm,the difference is only that AECM1 updates the degrees of freedom by maximizing the observed data likelihood function,while AECM2 updates the degrees of freedom by maximizing the expected complete data likelihood function.Experiments on a number of simulation data set show that the four maximum likelihood estimation algorithms ECME,ECM,AECM1 and AECM2 are insensitive to initial values.When the data contains outliers,the t BPPCA model is more accurate than the BPCCA model.Experiments on the face data set show that t BPPCA and BPCCA models have little difference in performance,however,when the data contains outliers,the error rate of t BPPCA is significantly lower than the BPCCA model.As the proportion of outliers increases,the error rate of BPPCA model increases obviously,but the t BPPCA model is almost unaffected by outliers.In contrast,t BPPCA is less sensitive to outliers and is more robust than the BPPCA model.