Tensor Decomposition And Regression Modeling Under The Neural Network And Bayesian Framework

As a multi-dimensional array,tensors are widely used in signal processing,image analysis,statistics and other fields.Its large-scale and special data characteristics have good applicability in mathematical analysis and are more comprehensive in data information extraction.In this thesis,under the framework of neural network and Bayesian,the theoretical research and practical application of tensors in parameter estimation,model construction,and data mining are considered.For the decomposition of tensors,similar to matrix decomposition,tensors are decomposed into the form of outer products with vectors as the component factors,and the resulting problem of minimal approximation error has been widely studied in the theoretical circle.For tensor decomposition,consider using spatial neural network,reasonably construct the objective function to train the decomposition components,and make the approximation error as small as possible.The results of the thesis show that the neural network algorithm is equivalent to the iterative least squares algorithm under the traditional decomposition.Secondly,under the linear model,combined with the Bayesian framework,the tensor parameter estimation problem is considered,compared with the traditional estimation method,a posterior estimation step of the tensor is proposed.According to the idea of bimodal Gibbs sampling in Ishwaran and Rao’s article,aiming at the complex data structure,we deduce the posterior of relevant parameters under the more abundant prior,and establish the Gibbs sampling process.For the regression modeling task of tensor data,the Bayesian neural network model is considered here.The parameter feedforward feedback mechanism in the traditional neural network model is combined with the prior and posterior action principle under Bayesian so that the parameter is no longer fixed.The purpose of this processing is to add a priori to enrich the parameter information,and to improve the convergence speed slightly.In the simulation,compared with the BP neural network,the results show that the performance of the Bayesian neural network model is stable and the results are good.In this thesis,the regression analysis of PM2.5 is carried out on the air quality data,and the results show that the model has a certain fitting ability.In general,the simulation experiments and case studies involved in the abovementioned research contents,the advantages of using tensor structural data are analyzed,and the feasibility and rationality of theoretical analysis are proved.This project clarifies the structural characteristics of tensors by analyzing neural networks and introducing a Bayesian framework.The main results are as follows: Different solutions for tensor decomposition and regression are proposed,and the other is to do some basic explorations in the theory and application of deep learning to provide a certain direction for its unification into statistical machine learning.