Research And Exploration On Randomized Algorithms For The Approximations Of Tensor Tucker Decompositions

Tensor Tucker decomposition algorithm plays an important role in high-dimensional data processing and other practical applications,and the accurate calculation of Tucker decomposition requires high memory and time.In many practical applications,approxi-mate Tucker decomposition with certain errors can meet the requirements,In this paper,we try to use two kinds of random matrices(symmetric Bernoulli random matrix and SRTT random matrix)to construct the random algorithm of tensor approximate Tucker decomposition,give the experimental data,and analyze the error upper bound of these two kinds of matrices Tensor Tucker decomposition algorithm plays an important role in high-dimensional data processing and other practical applications,and the accurate calculation of Tucker decomposition requires high memory and time.In many practi-cal applications,the approximate Tucker decomposition with some errors can meet the requirements.In this paper,we use the idea of random factorization in matrix for ref-erence,and extend it to tensor.We try to use two kinds of random matrices(symmetric Bernoulli random matrix and SRTT random matrix)to construct the randomized algo-rithm for the approximations of tensor Tucker decomposition,and analyze the upper bound of error of these two kinds of matrices.The main contents of this paper are as follows:(1)The symmetric Bernoulli random matrix is applied to the stochastic algorithm of tensor approximate Tucker decomposition.This paper analyzes how the stochastic algorithm of tensor approximate Tucker decomposition is obtained from the original form of matrix decomposition.Numerical experiments show that the experimental ef-fect is similar to that of Gaussian random matrix,but the symmetric Bernoulli random matrix is simple,Matrix elements can only take-1 or 1,so in the computer implemen-tation,by operating the bits of symbols in floating-point memory,it is possible to reduce the floating-point multiplication operation,and then make the calculation process more efficient than the Gaussian random matrix using floating-point numbers(2)In this paper,the theoretical error analysis of the spectral norm of the random-ized algorithm for the approximations of tensor Tucker decomposition using symmetric Bernoulli random matrix is carried out.It is proved that the symmetric Bernoulli ran-dom variable satisfies the sub-Gaussian and isotropic properties,and then the theoretical error of the spectral norm of the randomized algorithm for the approximations of tensor Tucker decomposition using symmetric Bernoulli random matrix is obtained.(3)The SRTT matrix,which is used to accelerate the process of stochastic decom-position,is applied to the stochastic algorithm of tensor approximate Tucker decompo-sition,and the numerical experiments and error analysis are carried out.