Nonnegative approximations of nonnegative tensors (0903.4530v2)

Abstract: We study the decomposition of a nonnegative tensor into a minimal sum of outer product of nonnegative vectors and the associated parsimonious naive Bayes probabilistic model. We show that the corresponding approximation problem, which is central to nonnegative PARAFAC, will always have optimal solutions. The result holds for any choice of norms and, under a mild assumption, even Bregman divergences.

• The paper proves that a best nonnegative rank-r approximation always exists for any nonnegative tensor, addressing a limitation in general tensor decomposition.
• It extends nonnegative matrix factorization (NMF) to higher-order tensors, defines nonnegative rank, and shows nonnegative approximation avoids PARAFAC degeneracy.
• This work guarantees the existence of optimal solutions, improving the reliability and convergence of algorithms for nonnegative tensor factorization in diverse applications.

Nonnegative Approximations of Nonnegative Tensors

The paper by Lek-Heng Lim and Pierre Comon explores a critical issue in computational mathematics: the decomposition of nonnegative tensors into sums of outer products of nonnegative vectors. This problem is central to the paper of nonnegative CANDECOMP/PARAFAC (CP) models, which extend the concept of matrix factorization to higher-dimensional data structures called tensors. The authors tackle foundational questions about the existence of optimal approximations of nonnegative tensors, shedding light on a challenging issue that emerges when dealing with tensors of order three or higher.

Key Findings

The paper establishes several essential results regarding nonnegative tensor approximations:

1. Existence of Optimal Solutions: Lim and Comon prove that for any nonnegative tensor, a best nonnegative rank-r approximation exists. This is significant because, for general tensors, such optimal solutions do not always exist when k (tensor order) is three or higher.
2. Generalization of Matrix Factorization: The paper extends the nonnegative matrix factorization (NMF) framework to tensors, resulting in the nonnegative PARAFAC model, also known as nonnegative tensor factorization (NTF). The solution to the tensor approximation problem always exists when nonnegativity constraints are imposed.
3. Proposition of Nonnegative Rank: The paper provides a formal definition of the nonnegative rank of a tensor, which is the minimum number of rank-1 nonnegative tensors required to express the given tensor as a sum.
4. Comparison with Classical Approaches: The work elucidates how the nonnegative approximation of tensors fundamentally differs from classical CP models in that it avoids the issue of 'PARAFAC degeneracy,' a situation where solutions can become ill-conditioned.

Implications

Theoretical Implications

The research advances the understanding of tensor decompositions by establishing conditions under which optimal solutions exist for nonnegative tensor approximations. The notion of nonnegative rank offers a new perspective, analogous to the rank of matrices, but tailored for the specific constraints of nonnegativity. This insight is particularly relevant for applications in areas that generate inherently nonnegative data matrices, such as chemometrics and probabilistic latent semantic indexing (pLSI) in natural language processing.

Practical Implications

The practical applications of this research are vast, impacting fields that rely on modeling multi-way data like signal processing, computer vision, and bioinformatics. The assurance of the existence of optimal approximations means that algorithms developed based on nonnegative tensor factorization can be expected to converge to meaningful solutions in finite time, thereby improving reliability and performance across various applications.

Future Directions

This paper lays a strong foundation for further investigations into more complex aspects of tensor decompositions. One potential avenue for future research is the exploration of algorithmic solutions optimized for computational efficiency, especially given the assurance of convergence to optimal solutions. Additionally, the relationship between different measures of proximity, such as Bregman divergences, and nonnegative tensor approximations is an area ripe for exploration. These divergences offer possibilities for interpretations related to information theory, potentially broadening the applicability of nonnegative tensor methodologies.

Conclusion

In summary, Lim and Comon have made substantial progress in the field of tensor approximations by demonstrating that nonnegative tensor factorizations always yield optimal solutions. This work not only addresses the pressing issue of solution existence in higher-order tensor approximations but also solidifies the theoretical underpinnings necessary for the practical implementation of these mathematical models in diverse scientific disciplines.