Primal-Dual Algorithms for Non-negative Matrix Factorization with the Kullback-Leibler Divergence

Non-negative matrix factorization (NMF) approximates a given matrix as a product of two non-negative matrices. Multiplicative algorithms deliver reliable results, but they show slow convergence for high-dimensional data and may be stuck away from local minima. Gradient descent methods have better behavior, but only apply to smooth losses such as the least-squares loss. In this article, we propose a first-order primal-dual algorithm for non-negative decomposition problems (where one factor is fixed) with the KL divergence, based on the Chambolle-Pock algorithm. All required computations may be obtained in closed form and we provide an efficient heuristic way to select step-sizes. By using alternating optimization, our algorithm readily extends to NMF and, on synthetic examples, face recognition or music source separation datasets, it is either faster than existing algorithms, or leads to improved local optima, or both

Culture and Human Capital Investments: Evidence of an Unconditional Cash Transfer Program in Bolivia

This paper uses a policy quasi-experiment created by the introduction of an old-age unconditional cash transfer program in Bolivia to study the intra-household income allocation process towards children's educational expenditure by ethnicity and gender of the recipient. Taking advantage of a sharp discontinuity created by the program assignment mechanism, I investigate the heterogeneity in the patterns of allocation within indigenous, multiethnic, and non-indigenous families, conditional on having one elder and one school-age child living in the household. I find that cultural factors (proxied by ethnicity) count in the decision making process of human capital investments. In particular, the allocation of resources within indigenous families follows rules closely related to patriarchal family structures (in which women have limited decision-making power) and is consistent with unitary, dictatorial, and common preferences theoretical household models. Conversely, non-indigenous families follow decision rules more closely related to collective and bargaining behavior models.Bolivia, culture, Bolivida, educational expenditure