Quantum Divide-and-Conquer Anchoring for Separable Non-negative Matrix Factorization

It is NP-complete to find non-negative factors W and H with fixed rank r from a non-negative matrix X by minimizing ||X-WH^Τ ||^2. Although the separability assumption (all data points are in the conical hull of the extreme rows) enables polynomial-time algorithms, the computational cost is not affordable for big data. This paper investigates how the power of quantum computation can be capitalized to solve the non-negative matrix factorization with the separability assumption (SNMF) by devising a quantum algorithm based on the divide-and-conquer anchoring (DCA) scheme [Zhou et al., 2013]. The design of quantum DCA (QDCA) is challenging. In the divide step, the random projections in DCA is completed by a quantum algorithm for linear operations, which achieves the exponential speedup. We then devise a heuristic post-selection procedure which extracts the information of anchors stored in the quantum states efficiently. Under a plausible assumption, QDCA performs efficiently, achieves the quantum speedup, and is beneficial for high dimensional problems.

ANALYZING LABOR HETEROGENEITY IN GHANAIAN COCOA PRODUCTION AND ITS IMPLICATIONS FOR SEPARABILITY IN HOUSEHOLD DECISIONS AND POLICY ASSESSMENT

Researchers have employed farm household models (FHMs) for policy analysis under the separability assumption. However, separability can fail, and the household's production and consumption decisions become simultaneous. Using 5 years of household data, the separability assumption among Ghana's cocoa-producing households is tested via heterogeneity of household adult males and females, household children, and hired and exchange labor. Results show labor is heterogeneous, implying a lack of separability. Simulation analysis also shows that ignoring nonseparability leads to an underestimation of policy effects. Thus, nonseparability in production and consumption decisions must be incorporated in FHMs developed for Ghanaian cocoa-producing households.

Matching by Categories

This chapter considers matching by categories, beginning with a discussion of a specific but empirically very relevant family of models in order to provide a richer representation of heterogeneity between individuals that would account not only for economic aspects such as income or education, as well as more subjective (and less easily observable) ones, such as idiosyncratic preferences for marriage in general and for specific types of spouses in particular. The chapter explores a simple model that describes matching on income or education, the separability assumption, how separability can be justified, and the dual structure under separability. It also provides an overview of the Choo-Siow model, focusing on its basic structure, the matching function, heteroskedasticity, comparative statics, testability and identifiability, and the Galichon and Salanié's cupid framework as an extension of the model.