All-at-once Optimization for Coupled Matrix and Tensor Factorizations (1105.3422v1)

Abstract: Joint analysis of data from multiple sources has the potential to improve our understanding of the underlying structures in complex data sets. For instance, in restaurant recommendation systems, recommendations can be based on rating histories of customers. In addition to rating histories, customers' social networks (e.g., Facebook friendships) and restaurant categories information (e.g., Thai or Italian) can also be used to make better recommendations. The task of fusing data, however, is challenging since data sets can be incomplete and heterogeneous, i.e., data consist of both matrices, e.g., the person by person social network matrix or the restaurant by category matrix, and higher-order tensors, e.g., the "ratings" tensor of the form restaurant by meal by person. In this paper, we are particularly interested in fusing data sets with the goal of capturing their underlying latent structures. We formulate this problem as a coupled matrix and tensor factorization (CMTF) problem where heterogeneous data sets are modeled by fitting outer-product models to higher-order tensors and matrices in a coupled manner. Unlike traditional approaches solving this problem using alternating algorithms, we propose an all-at-once optimization approach called CMTF-OPT (CMTF-OPTimization), which is a gradient-based optimization approach for joint analysis of matrices and higher-order tensors. We also extend the algorithm to handle coupled incomplete data sets. Using numerical experiments, we demonstrate that the proposed all-at-once approach is more accurate than the alternating least squares approach.

• The paper introduces CMTF-OPT, which simultaneously optimizes factor matrices for coupled matrix and tensor factorization to better capture latent structures.
• It extends the approach to handle missing data using weighted least squares, ensuring robust performance on incomplete datasets.
• Numerical experiments demonstrate that CMTF-OPT outperforms alternating least squares, especially under overfactoring and noise.

An Overview of "All-at-once Optimization for Coupled Matrix and Tensor Factorizations"

The paper "All-at-once Optimization for Coupled Matrix and Tensor Factorizations" addresses the challenges associated with jointly analyzing heterogeneous data sets—specifically, data sets comprised of both matrices and higher-order tensors. The authors propose a novel approach, termed CMTF-OPT (Coupled Matrix and Tensor Factorization Optimization), that advances beyond traditional alternating algorithms by optimizing all factor matrices simultaneously.

Technical Contributions

The CMTF-OPT algorithm is significant in its ability to jointly factor matrices and tensors through an all-at-once optimization method, employing a gradient-based approach. The main contributions of the paper can be highlighted as:

1. Simultaneous Optimization: Unlike traditional methods that solve for one factor matrix at a time, the CMTF-OPT leverages gradient-based optimization to handle multiple factor matrices and tensors collectively. This method aims to better capture the latent structure of the data.
2. Handling Missing Data: The paper extends CMTF-OPT to accommodate incomplete data sets, which is crucial given real-world data's often fragmentary nature. This extension makes use of weighted least squares to focus only on available data entries.
3. Numerical Superiority: The numerical experiments show that CMTF-OPT outperforms the alternating least squares (ALS) approach in terms of accuracy, especially when there is overfactoring—extracting more components than the true underlying factors.

Key Findings

The paper extensively evaluates CMTF-OPT against ALS-based methods through simulations involving randomly generated data. The findings show that:

• CMTF-OPT demonstrates greater robustness to overfactoring, maintaining high accuracy where ALS tends to fail.
• It consistently delivers superior factor recovery when fitting the correct number of components.
• CMTF-OPT exhibits greater resilience to noise, preserving accuracy across varying noise levels.

Implications and Future Directions

The methodology introduced in the paper holds substantial implications for domains that require data fusion from varied sources, such as recommendation systems and medical diagnostics. The practice of optimizing coupled matrices and tensors simultaneously can enhance the interpretability and predictive power of models in these contexts.

In terms of future developments, the paper suggests exploration into different loss functions that could extend CMTF-OPT to various noise types and data configurations. Moreover, incorporating constraints such as nonnegativity could aid in improving the interpretability of factor matrices. The authors also recognize the potential in applying Bayesian frameworks to the proposed optimization problem, which might address current limitations regarding scale ambiguities in some data factorizations.

Conclusion

This work contributes significantly to the field of data mining and machine learning by presenting a robust and scalable method for analyzing heterogeneous data. CMTF-OPT opens new avenues for multi-modal data processing, emphasizing accuracy and reliability over existing methods. Its adaptability to missing data scenarios further accentuates its applicability across diverse applications in industry and academia.