An iterative algorithm for solving ill-conditioned linear least squares problems

Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm（SAIA） is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.

LapRLSR for NIR spectral modeling and its application to online monitoring of the column separation of Salvianolate

A novel near infrared （NIR） modeling method-Laplacian regularized least squares regression （LapRLSR） was presented,which can take the advantage of many unlabeled spectra to promote the prediction performance of the model even if there are only few calibration samples. Using LapRLSR modeling, NIR spectral analysis was applied to the online monitoring of the concentration of salvia acid B in the column separation of Salvianolate. The results demonstrated that LapRLSR outperformed partial least squares （PLS） significantly, and NIR online analysis was applicable.

A Simple Hybrid Recursive Learning Algorithm with High Generalization Performance for Radial Basis Function Neural Network

In this paper, we propose a simple learning algorithm for non\|linear function approximation and system modeling using minimal radial basis function neural network with high generalization performance. A hybrid algorithm is constructed, which combines recursive n \|means clustering algorithm with a simple recursive regularized least squares algorithm (SRRLS). The n \|means clustering algorithm adjusts the centers of the network, while the SRRLS constructs a parsimonious network which makes the generalization performance of the network well. The SRRLS algorithm needs no matrix computing, so it has a lower computational cost and no ill\|conditional problem. Because of the recursive manner, this algorithm is suitable for on\|line applications. The effectiveness of this algorithm is demonstrated by two benchmark examples.

Non-asymptotic Error Bound for Optimal Prediction of Function-on-Function Regression by RKHS Approach

In this paper,we study and analyze the regularized least squares for function-on-function regression model.In our model,both the predictors(input data)and responses(output data)are multivariate functions(with d variables andd variables respectively),and the model coefficient lies in a reproducing kernel Hilbert space(RKHS).We show under mild condition on the reproducing kernel and input data statistics that the convergence rate of excess prediction risk by the regularized least squares is minimax optimal.Numerical examples based on medical image analysis and atmospheric point spread function estimation are considered and tested,and the results demonstrate that the performance of the proposed model is comparable with that of other testing methods.