Research On Some Problems Of Random Matrix Theory And Machine Learning

Random matrix theory mainly studies the statistical properties of eigenvalues(or singular values)of random matrices.It has grown into a vital area of probability and played an important role in many fields including machine learning,operational research and cybernetics,and com-putational mathematics.In Chapter 3 and 4,we study the deviation inequalities of eigenvalues(or singular values)of random matrices.With the development of artificial intelligence,the complex data generated in practical industrial process can be effectively analyzed based on machine learning methods,and the ana-lyzing results can in turn provide a powerful guidance to the efficiency improvement and the cost control for the industrial production.Chapter 5 mainly focuses on the application of machine learning in tunnel boring machines(TBM).Chapter 3 is divided into three parts to study the large deviation inequality of the largest eigenvalue(or singular values)of random matrices.In the first part,we study concentra-tion inequalities of the largest eigenvalue of a matrix infinitely divisible(i.d.)series,which is a finite sum of fixed matrices weighted by i.d.random variables.By using matrix moment generating function for i.d.distribution,we obtain several types of tail inequalities,including Bennett-type and Bernstein-type inequalities.This allows us to further bound the expectation of the spectral norm of a matrix i.d.series.Moreover,by developing a piecewise function for Q(s)=(s+1)log(s+1)-s that appears in the Bennett-type inequality,we derive a tighter con-centration inequality of the largest eigenvalue of the matrix i.d.series than the Bernstein-type inequality when the matrix dimension is high.The resulting lower-bound function can improve any Bennett-type concentration inequality that involves the function Q(s).In the second part,we present the dimension-free concentration inequalities and expecta-tion bounds for the largest singular value(LS V)of matrix Gaussian series,respectively.Among them,the expected upper bound is obtained by the concentration inequality and direct deriva-tion.By using the resulting bounds,we compute the concentration inequalities and expectation bounds for LS Vs of Gaussian Wigner matrix and symmetrical Gaussian Toeplitz matrix,respec-tively.In the third part,we obtain a concentration inequality for the largest singular value of the sub-Gaussian matrix.Two methods are used to obtain the results,one is to convert the largest singular value of sub-Gaussian random matrix into sub-Gaussian variables,and the other is to convert the sub-Gaussian matrix into sub-Gaussian matrix series.The combination of two results leads to the final conclusion.As an application,we use the resulting theorem to compute the concentration inequality of the Gaussian Toeplitz matrix.In Chapter 4,we present the small deviation inequalities for the largest eigenvalues of sums of random matrices.In particular,we first present some basic small deviation results of random matrices.We then obtain several types of small deviation inequalities for the largest eigenvalue of sums of independent random positive semi-definite(PSD)matrices.The resulting small deviation inequalities are independent of the matrix dimension and thus our finding are applicable to the high-dimensional and even infinite-dimensional cases.In Chapter 5,we introduce several machine learning models and use in-situ operating data to deal with the prediction of TBM operating parameters,including the torque,the velocity,the thrust and the chamber pressure.We use three kinds of recurrent neural networks(RNN),including traditional RNN,long-short term memory(LSTM)networks and gated recurrent unit(GRU)networks.We also make a comparison with several shallow models,such as support vector regression(SVR),random forest(RF)and Lasso.The experimental results show that the proposed RNN-based predictors outperform the shallow models in most cases.The feasibility of RNN for the real-time prediction of TBM operating parameters indicates that RNN can afford the analysis and the forecasting of the time-continuous various construction equipments.