Member-only story
Reinforce Matrix Factorization for Time Series Modeling: Probabilistic Sequential Matrix Factorization
How to integrate state-space model into matrix factorization in a probabilistic framework?
Matrix factorization is a classical machine learning approach for modeling real-world data. It shows great potential for solving various data reconstruction and imputation problems such as recommender system and image inpainting. Recently, matrix factorization has also become an efficient tool for modeling real-world time series data. Since time series data involve strong temporal dependencies, matrix factorization variants are required to achieve temporal/sequential modeling. Today, we will introduce a probabilistic sequential matrix factorization (PSMF) model to high-dimensional time series modeling. The model takes into account nonlinear Gaussian state-space models for achieving temporal nonlinearities. Notably, PSMF model was proposed in the following paper:
Akyildiz, O. D., van den Burg, G., Damoulas, T., & Steel, M. (2021). Probabilistic sequential matrix factorization. In International Conference on Artificial Intelligence and Statistics (AISTATS 2021) (pp. 3484–3492). PMLR.
- Slides: https://probnum2022.github.io/pdf/akyildiz.pdf
- Python implementation: https://github.com/alan-turing-institute/rPSMF
Matrix Factorization
Matrix factorization is a simple yet efficient approach for many real-world data modeling problems. Given any data matrix Y of size d-by-n, we can build a matrix factorization problem as follows,
In such approximation problem, as we define a low rank as r (positive integer), d-by-r matrix C is the dictionary matrix, and r-by-n matrix X is the coefficient matrix. In practice, one important application of matrix factorization is dimensionality reduction — seeking low-rank representation and interpretability. Going back to the above matrix factorization formula, we aim at learning the dictionary matrix and the coefficient matrix from given data matrix. Sometimes, if the data matrix is partially observed, we can also utilize matrix factorization to reconstruct missing values.
