Computing a Nonnegative Matrix Factorization -- Provably (1111.0952v1)

Abstract: In the Nonnegative Matrix Factorization (NMF) problem we are given an $n \times m$ nonnegative matrix $M$ and an integer $r > 0$. Our goal is to express $M$ as $A W$ where $A$ and $W$ are nonnegative matrices of size $n \times r$ and $r \times m$ respectively. In some applications, it makes sense to ask instead for the product $AW$ to approximate $M$ -- i.e. (approximately) minimize $\norm{M - AW}_F$ where $\norm{}_F$ denotes the Frobenius norm; we refer to this as Approximate NMF. This problem has a rich history spanning quantum mechanics, probability theory, data analysis, polyhedral combinatorics, communication complexity, demography, chemometrics, etc. In the past decade NMF has become enormously popular in machine learning, where $A$ and $W$ are computed using a variety of local search heuristics. Vavasis proved that this problem is NP-complete. We initiate a study of when this problem is solvable in polynomial time: 1. We give a polynomial-time algorithm for exact and approximate NMF for every constant $r$. Indeed NMF is most interesting in applications precisely when $r$ is small. 2. We complement this with a hardness result, that if exact NMF can be solved in time $(nm)^{o(r)}$, 3-SAT has a sub-exponential time algorithm. This rules out substantial improvements to the above algorithm. 3. We give an algorithm that runs in time polynomial in $n$, $m$ and $r$ under the separablity condition identified by Donoho and Stodden in 2003. The algorithm may be practical since it is simple and noise tolerant (under benign assumptions). Separability is believed to hold in many practical settings. To the best of our knowledge, this last result is the first example of a polynomial-time algorithm that provably works under a non-trivial condition on the input and we believe that this will be an interesting and important direction for future work.

• The paper introduces a polynomial-time algorithm for exact and approximate NMF when the factorization rank is constant.
• The paper proves that significant improvements would imply sub-exponential time for 3-SAT, highlighting inherent computational limits.
• The paper develops an efficient, noise-robust algorithm under the separability condition, enhancing NMF's practical application in data analysis.

Nonnegative Matrix Factorization: Algorithmic Developments and Theoretical Insights

The paper "Computing a Nonnegative Matrix Factorization — Provably" by Sanjeev Arora et al. addresses the Nonnegative Matrix Factorization (NMF) problem, an important problem with wide-ranging applications in fields like data analysis, quantum mechanics, and machine learning among others. The primary goal in NMF is to decompose a given nonnegative matrix $M$ into the product of two nonnegative matrices $A$ and $W$. This decomposition has garnered interest due to its applicability in uncovering latent relationships particularly when $r$, the inner dimension of the factorization, is small.

Summary of Contributions

The authors make several key contributions toward understanding NMF both from a computational complexity perspective and a practical algorithmic approach:

1. Polynomial-Time Algorithm for Small $r$: The paper introduces a polynomial-time algorithm for both exact and approximate NMF when $r$ is constant. This is significant as NMF is most applicable and insightful in scenarios where $r$ is small, such as in document clustering where $r$ represents a reduced set of topics.
2. Hardness Results: Complementing the algorithmic development, the paper provides a hardness result showing that if exact NMF can be solved in time $(nm)^{o(r)}$, this would imply a sub-exponential time algorithm for the $3$-SAT problem, thereby suggesting that substantial improvements over the presented algorithm are unlikely.
3. Algorithm Under Separability: For scenarios where the input matrix satisfies the separability condition, the authors provide an efficient algorithm that is robust to noise and practical for implementations requiring polynomial time with respect to $n$, $m$, and $r$. This condition is believed to hold in many practical settings and represents a significant reduction in computational complexity in these cases.

Theoretical and Practical Implications

The theoretical implications of this work underscore the intrinsic difficulty of the NMF problem, paralleling the complexity found in fundamental problems like $3$-SAT. The results serve to guide future research by identifying constraints like separability that can make the problem tractable. Practically, the algorithms provided, especially under separability conditions, have the potential to improve NMF application in real-world scenarios such as image recognition, natural language processing, and bioinformatics where computational efficiency is paramount.

Speculations on Future Developments

Given the landscape painted by this research, future directions could focus on exploring broader classes of conditions under which the complexity of NMF is reduced. Additionally, further developments may include refining algorithms to handle noise more effectively or extending separability-type assumptions in contexts like deep learning models where interpretability and efficiency are critical.

Overall, the paper not only furthers our understanding of the computational complexity of NMF but also provides practical tools that can bridge the gap between theoretical advancement and applied data science and machine learning frameworks.