On the complexity of nonnegative matrix factorization (0708.4149v2)

Abstract: Nonnegative matrix factorization (NMF) has become a prominent technique for the analysis of image databases, text databases and other information retrieval and clustering applications. In this report, we define an exact version of NMF. Then we establish several results about exact NMF: (1) that it is equivalent to a problem in polyhedral combinatorics; (2) that it is NP-hard; and (3) that a polynomial-time local search heuristic exists.

• The paper proves that exact NMF is NP-hard by establishing its equivalence to the INTERMEDIATE SIMPLEX problem in polyhedral combinatorics.
• The study introduces a polynomial-time local search heuristic that offers a practical approach to approximating NMF solutions on large datasets.
• The reduction to INTERMEDIATE SIMPLEX connects NMF with established polyhedral methods, paving the way for new algorithmic developments in data analysis and clustering.

On the Complexity of Nonnegative Matrix Factorization

The paper by Stephen A. Vavasis explores the computational complexity of the Nonnegative Matrix Factorization (NMF) problem, providing fundamental insights into its nature and implications for various applications in data analysis and clustering.

Key Contributions

The author delineates the complexities associated with a specific variant of NMF, termed "exact NMF," which requires the decomposition of a given nonnegative matrix $A$ into two nonnegative matrices $W$ and $H$ such that $A = WH$. The paper's primary contributions are as follows:

1. Equivalence to Polyhedral Combinatorics: The paper establishes the equivalence of exact NMF to a problem in polyhedral combinatorics, termed INTERMEDIATE SIMPLEX. This equivalence provides a new perspective on the challenge of achieving exact NMF solutions.
2. NP-Hardness of Exact NMF: By proving that exact NMF is NP-hard, the paper addresses a significant gap in the literature, indicating that there is no known polynomial-time algorithm to solve the problem optimally. This insight critically informs the practical approaches to solving NMF, highlighting the need for heuristic or approximation algorithms.
3. Local Search Heuristic: Despite the NP-hardness, a polynomial-time local search heuristic for NMF is demonstrated. This development offers a feasible approach to obtaining approximate solutions in practice, providing a valuable tool for applications in large datasets.

Detailed Analysis

Theoretical underpinnings are rigorously discussed, particularly the use of polyhedral combinatorics to delineate the boundaries of exact NMF. The reduction to the INTERMEDIATE SIMPLEX problem and its subsequent proof of NP-hardness involves intricate polynomial-time transformations and arithmetic operations.

The paper provides proofs through polynomial-time reductions between exact NMF and RESTRICTED P1, leading to the INTERMEDIATE SIMPLEX problem, thereby laying down a structurally robust pathway from NMF to polyhedral problems. This methodology affirms prior propositions by Cohen and Rothblum on the NP-hard nature of nonnegative rank determination and extends these results.

Implications and Future Directions

The establishment of NP-hardness for exact NMF has profound implications:

• Algorithmic Development: Given the impracticality of achieving exact solutions in reasonable timeframes for large instances, future work in algorithm development will likely focus on improving heuristic and approximation methods.
• Theoretical Exploration: The equivalence to polyhedral combinatorics suggests potential new avenues for revisiting and reformulating existing problems within computational geometry and linear programming frameworks.
• Practical Applications: Areas such as image and text data clustering and feature extraction will benefit from more sophisticated approximation techniques that can handle the confirmed complexity.

In conclusion, Vavasis’s exploration of the NMF problem provides critical insights into its complexity, guiding both theoretical research and practical algorithm development. The work underscores the need for ongoing innovation in heuristic methodologies to effectively leverage NMF in data-intensive environments. Future research may build upon this foundational framework to explore more efficient ways of navigating the problem’s inherent computational challenges.