Research On Probabilistic Tri-factorization Algorithms Of Matrics

With the development of the information age,we are facing the challenge of processing and analysing massive high-dimensional data.How to obtain low-rank structure of data is the key to meet such chal-lenges.Low-rank matrix decomposition is considered as a method to extract the low-rank structure,to remove the effect of noise or to fill in the missing values,and enhance the efficiency of the algorithm.In this paper,we mainly consider probabilistic matrix tri-factorization model.Compared to traditional matrix tri-factorization algorithm,the algorithms based on probabilistic models performs better in express-ing the noise of the observation data,can randomly generate the ob-servation data under the condition of given some parameters,and can implement the statistic inference.In order to alleviate the effect of outliers in the matrix tri-factorization algorithm,in this paper,we first construct a probabilistic matrix tri-factorization model combined with the distribution prior(TBMTF).In such a model,we assumes that the error term is subject to a univariate distribution and the potential characteristics are follow to a multivariate distribution with the variational bayesian inference being used to estimate the model parameters.Compared to the nor-mal distribution in the classical probabilistic matrix tri-factorization model,the distribution has a richer variation in the density curve.Especially,for the data with some extreme values or obvious fluctu-ations,the prior model based on the distribution presents better ro-bustness for outliers.It is shown based on a artificial data and one real dataset that the advantages of the TBMTF algorithm in analyzing the data with outlier.Notice that the negative elements may appear in the result of tra-dition matrix decomposition.However,in some real life problem,negative values are meaningless.In addition,since sparsity reduces the dimensionality of the data,the dependence between dimensions of the feature vector becomes lower after sparse representation.Thus,in this paper,we further consider the robust probability matrix de-composition model under non-negative constraints and sparse con-straints,and propose a probabilistic sparse non-negative matrix tri-factorization model(PSNMTF).In such a model,each value of the component matrices and (1 is given a truncated normal distribution prior of different parameters,and the precision of the truncated nor-mal distribution is adjusted by the gamma distribution to achieve the purpose of robustness,non-negativity and sparsity.Similarly,in this paper,we apply the variational bayesian inference to evaluate the pa-rameters in the PSNMTF model.In numerical examples,we verifies the effectiveness of the PSNMTF algorithm for image reconstruction,missing value prediction and noise detection on four real datasets,Yale face data,JAFFE face data,toy data and GDSC genetic data.