Economic Complexity and Matrix Factorization: Inferring Hidden Capabilities in Municipal Production Networks

In the framework of Economic Complexity, the concept of capabilities represents a fundamental theoretical construct for modeling economic systems. These capabilities, while extensively theorized as the hidden endowments driving countries' productive structures, have remained empirically elusive, never subjected to direct inference attempts. The present work addresses this critical gap by developing an approach to infer and extract information about this theoretical capabilities layer directly from empirical data. Our methodology employs Italian municipal-level economic data, specifically utilizing ATECO codes (the Italian classification system for economic activities harmonized with the European NACE nomenclature) organized in a binary matrix format, representing the presence or absence of economic activities in municipalities. We approach the problem through the lens of matrix factorization, treating the reconstruction of the municipality-activity matrix as an optimization problem where latent factors correspond to underlying economic capabilities. Employing a mask-and-predict methodology that systematically hide portions of the matrix to evaluate reconstruction accuracy, we determined that five capabilities constitute an excellent candidate in order to balance reconstruction precision against model parsimony. This dimensionality provides reconstruction accuracies already exceeding eighty percent for both the zeros and ones in the original binary matrix, demonstrating the method's robustness in capturing the underlying economic structure. The analysis reveals that the five identified degrees of freedom exhibit strong correlations with empirically observable economic-geographic properties of Italian municipalities, suggesting that our latent factors capture meaningful economic dimensions. Incorporating the fitness-complexity metrics into our model served dual purposes: consolidating the reliability of the Fitness-Complexity classification, which proved consistent with our analysis and revealing that the structural skeleton introduced by the Fitness-Complexity algorithm remains visible even when additional degrees of freedom are introduced. Specifically, reconstructions with more than one capability maintain a fundamental structure reminiscent of the Fitness-Complexity framework, suggesting that the approach captures essential features of economic organization that persist across different levels of dimensional reduction. Through appropriate null models, we demonstrated that the municipality-activity matrix possesses a sub-structure beyond what is captured by the fitness-complexity algorithm alone. This finding presents new challenges and opportunities for developing algorithms capable of revealing these deeper structural patterns. This work contributes to the economic complexity literature by providing the first systematic attempt at capabilities inference from empirical data, trying to bridge the gap between theoretical frameworks and observable economic patterns.