A review of mathematical approaches to recommendation algorithms: from collaborative filtering to deep learning

Recommendation algorithms are one of the most important technologies in modern information systems and are widely used in e-commerce, social networks, streaming systems and digital platforms. Their main goal is to provide personalized recommendations by analyzing user preferences and interactions. These systems are based on mathematical concepts from linear algebra, optimization theory, statistics and machine learning. This paper presents an overview of the most important mathematical models and recommendation algorithms, with a special emphasis on collaborative filtration, matrix factorization and modern approaches based on deep learning. Their theoretical foundations, advantages, limitations and evaluation criteria are analyzed. In addition, challenges related to data sparsity, the cold start problem and the computational complexity of the algorithms are considered. The paper provides a systematic overview of the development of recommender systems and identifies future research directions in this area.