Matrix factorization-based recommender system is in effect an angle preserving dimensionality reduction technique. Since the frequency of items follows power-law distribution, most vectors in the original dimension of user feature vectors and item feature vectors lie on the same hyperplane. However, it is very difficult to reconstruct the embeddings in the original dimension analytically, so we reformulate the original angle preserving dimensionality reduction problem into a distance preserving dimensionality reduction problem. We show that the geometric shape of input data of recommender system in its original higher dimension are distributed on co-centric circles with interesting properties, and design a paraboloid-based matrix factorization named ParaMat to solve the recommendation problem. In the experiment section, we compare our algorithm with 8 other algorithms and prove our new method is the most fair algorithm compared with modern day recommender systems such as ZeroMat and DotMat Hybrid.

Although recommender systems have received great success, We categorize recommender systems as shallow it is well known for highly skewed datasets, models and deep models. The first class engineers and researchers need to adjust their incorporates shallow machine learning technologies methods to tackle the specific problem to yield good such as matrix factorization and learning to rank, results. Inability to deal with highly skewed dataset while the second class are deep learning models like usually generates hard computational problems for Wide and Deep [6]. Although a bit of out-of-dated, big data clusters and unsatisfactory results for shallow models are still widely used in small customers. In this paper, we propose a new companies and projects where agility, usability and algorithm solving the problem in the framework of matrix factorization. We model the data skewness efficiency far outweighs boost of performance which factors in the theoretic modeling of the approach is only economically visible for huge datasets. It is with easy to interpret and easy to implement well known since the invention of the first shallow formulas. We prove in experiments our method model, that data skewness and sparsity poses generates comparably favorite results with popular serious challenges for recommender system recommender system algorithms such as Learning performance. The setbacks are two folds: data to Rank, Alternating Least Squares and Deep Matrix skewness causes problems that need special Factorization.

In real-world recommender systems, some users are easily influenced by new products and whereas others are unwilling to change their minds. So the preference varying speeds for users are different. Based on this observation, we propose a dynamic nonlinear matrix factorization model for collaborative filtering, aimed to improve the rating prediction performance as well as track the preference varying speeds for different users. We assume that user-preference changes smoothly over time, and the preference varying speeds for users are different. These two assumptions are incorporated into the proposed model as prior knowledge on user feature vectors, which can be learned efficiently by MAP estimation. The experimental results show that our method not only achieves state-of-the-art performance in the rating prediction task, but also provides an effective way to track user-preference varying speed.