# A Biased Proportional-Integral-Derivative-Incorporated Latent Factor Analysis Model
###### Abstract  
Nowadays, as the number of items is increasing and the number of items that users have access to is limited, user-item preference matrices in recommendation systems are always sparse. This leads to a data sparsity problem. The latent factor analysis (LFA) model has been proposed as the solution to the data sparsity problem. As the basis of the LFA model, the singular value decomposition (SVD) model, especially the biased SVD model, has great recommendation effects in high-dimensional sparse (HiDs) matrices. However, it has the disadvantage of requiring several iterations before convergence. Besides, the model PID-incorporated SGD-based LFA (PSL) introduces the principle of discrete PID controller into the stochastic gradient descent (SGD), the learning algorithm of the SVD model. It could solve the problem of slow convergence speed, but its accuracy of recommendation needs to be improved. In order to make better solution, this paper fuses the PSL model with the biased SVD model, hoping to obtain better recommendation result by combining their advantages and reconciling their disadvantages. The experiments show that this biased PSL model performs better than the traditional matrix factorization algorithms on different sizes of datasets.  
latent factor analysis; stochastic gradient descent; PID controller; biased SVD model +
Footnote †: journal: Applied  
19 May 2021  
15 June 2021  
**Published: 20 June 2021**  
situation, cold start [3], reduced coverage, neighbor transitivity and a series of problems need to be solved. Data sparsity [4] brings tons of difficulties to the recommendation. In other words, the user provides little information for reference by the recommendation system, but the number of items that are needed to predict the scores is quite large. In previous studies, scholars have proposed many solutions to these problems, such as singular value decomposition (SVD) [5], principal component analysis (PCA) [6], content-boosted CF algorithm (CBCF) [7], tree augmented naive bayes optimized by extended logistic regression (TAN-ELR) [8]. All of them have their pros and cons.  
Aiming to devise better solution to the data sparsity problem, this paper describes a recommendation model based on the latent factor analysis (LFA) model, which is a kind of model-based recommendation techniques of the collaborative filtering (CF) recommendation system [9]. It uses the idea of the biased SVD model to improve the PID-incorporated SGD-based LFA (PSL) model which combines the advantages of both of them. This model fuses the prediction method of biased SVD with the instantaneous error correction method of PSL, which solves the shortcomings of slow convergence of the biased SVD model and the problem of low recommendation accuracy of the PSL model. When compared with other recommendation models, experiments prove that it has high computational efficiency, as well as high prediction accuracy on the HiDS matrix. The contributions of this paper are as follows:  
1. It introduces a biased PSL model combining the biased SVD and the PSL model;
2. Experiments on three large datasets demonstrate that the biased PSL model can achieve highly competitive prediction accuracy for the missing data of an HiDS matrix compared to the other models.  
The rest of this paper is organized as follows: Section 2 shows the related work about the research; Section 3 gives the preliminaries knowledge for a detailed introduction to the basis of the algorithm. Section 4 describes the specific implementation process of the algorithm. Section 5 reports the recommendation results of the algorithm and compares it with previous recommendation methods; Section 6 evaluates the performance of each method; Finally, Section 7 summarizes the whole paper and puts forward the direction of future research.