Complexity of entrywise power matrix factorization

Matrix factorization is a fundamental tool in areas such as data analysis, recommendation systems, and signal processing. However, when nonlinear constraints are imposed, such as raising each entry to a fixed power, the problem acquires a computational complexity that deserves detailed attention. A recent article addresses the so-called entrywise power matrix factorization (EPMF), which seeks to decompose a nonnegative matrix into the product of two low-rank factors, subsequently applying component-wise exponentiation. This model generalizes known cases such as the modulus (p=1) and the square (p=2), the latter linked to the square root rank. Interestingly, the exact version of the problem reduces to deciding whether it is possible to alter the signs of the entries of a given matrix to obtain a fixed rank, a combinatorial challenge that researchers have shown to be strongly NP-hard in general, although it admits polynomial-time algorithms when the rank is constant. In the approximate case, measured with the Frobenius norm, difficulty arises even for rank 2, the minimal nontrivial case. These conclusions draw a complete complexity landscape that guides both academics and professionals who need to implement efficient solutions. In practice, understanding these limits is key to designing robust numerical methods and heuristics. Companies working with large volumes of data, such as those developing custom applications, benefit from knowing which algorithms are computationally feasible and which require approximations. At Q2BSTUDIO, we offer AI services for businesses that integrate optimization and machine learning techniques to solve complex factorization and dimensionality reduction problems. Additionally, we have specialized teams in cybersecurity, aws and azure cloud services, and business intelligence services with power bi, capable of deploying infrastructures that execute these models in a scalable manner. Research on the complexity of EPMF is not only a theoretical advance but also a practical guide for those developing custom software and AI agents that require performance guarantees. Knowing which instances are tractable allows for better design decisions, whether implementing exact solutions for fixed ranks or resorting to efficient approximate methods. At Q2BSTUDIO, we apply this knowledge to build systems that transform data into value, combining analytical depth with the robustness of our cloud and business intelligence platforms.