Mayank Garg,
- Student, Department of Artificial Intelligence and Data Science ADGIPS, New Delhi, India
Abstract
Matrix factorization and tensor decomposition techniques have emerged as fundamental tools in machine learning and data science for handling high dimensional data efficiently. This paper presents a comprehensive analysis of scalable matrix factorization and tensor decomposition methods, focusing on their mathematical foundations, computational complexity, and practical applications. We examine key algorithms including Singular Value Decomposition (SVD), Non-negative Matrix Factorization (NMF), CP decomposition, and Tucker decomposition, with particular emphasis on their scalability challenges and solutions. Our analysis reveals that while traditional methods face computational bottlenecks with large-scale data, recent advances in randomized algorithms, distributed computing, and streaming approaches offer promising solutions. The paper provides detailed mathematical formulations, complexity analysis, and discusses applications in recommender systems, image processing, and data compression. Results indicate that tensor decomposition methods can achieve compression ratios of up to 90% while maintaining reconstruction accuracy above 95% for typical datasets.
Keywords: Matrix factorization, tensor decomposition, scalable algorithms, dimensionality reduction, machine learning, data compression, distributed computing.
[This article belongs to Recent Trends in Mathematics ]
References
- G. Kolda and B. W. Bader,” Tensor decompositions and applications, “SIAM Review, vol. 51, no. 3, pp. 455-500, 2009.
- Fan,” Multi-mode deep matrix and tensor factorization,” International Conference on Learning Representations, 2022.
- D. Carroll and J. J. Chang,” Analysis of individual differences in multidimensional scaling via an n-way generalization of Eckart-Young decomposition,” Psychometrika, vol. 35, pp. 283-319, 1970.
- D. Lee and H. S. Seung,” Learning the parts of objects by non-negative matrix factorization,” Nature, vol. 401, pp. 788-791, 1999.
- Halko, P. G. Martinsson, and J. A. Tropp,” Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions,” SIAM Review, vol. 53, no. 2, pp. 217-288, 2011.
- Symeonidis and A. Zioupos,” Matrix and tensor factorization techniques for recommender systems,” Springer, 2016.
- Koren, R. Bell, and C. Volinsky,” Matrix factorization techniques for recommender systems,” Computer, vol. 42, no. 8, pp. 30-37, 2009.
- Cichocki et al.,” Tensor decompositions for signal processing applications,” IEEE Signal Processing Magazine, vol. 32, no. 2, pp. 145-163, 2015.
- Rajih et al.,” Enhanced line search: A novel method to accelerate PARAFAC,” SIAM Journal on Matrix Analysis and Applications, vol. 30, no. 3, pp. 1128-1147, 2008.
- R. Tucker,” Some mathematical notes on three-mode factor analysis,” Psychometrika, vol. 31, pp. 279-311, 1966.
- A. Harshman,” Foundations of the PARAFAC procedure: Models and conditions for an explanatory multi-modal factor analysis,” UCLA Working Papers in Phonetics, vol. 16, pp. 1-84, 1970.
- Kim and H. Park,” Algorithms for nonnegative matrix and tensor factorizations: A unified view based on block coordinate descent framework,” Journal of Global Optimization, vol. 58, pp. 285-319, 2014.
Recent Trends in Mathematics
| Volume | 03 | |
| Issue | 02 | |
| Received | 14/03/2026 | |
| Accepted | 13/08/2026 | |
| Published | 28/08/2026 | |
| Publication Time | 167 Days |