Topological Data Analysis with Bregman Divergences

Given a finite set in a metric space, the topological analysis generalizes hierarchical clustering using a 1-parameter family of homology groups to quantify connectivity in all dimensions. The connectivity is compactly described by the persistence diagram. One limitation of the current framework is the reliance on metric distances, whereas in many practical applications objects are compared by non-metric dissimilarity measures. Examples are the Kullback-Leibler divergence, which is commonly used for comparing text and images, and the Itakura-Saito divergence, popular for speech and sound. These are two members of the broad family of dissimilarities called Bregman divergences. We show that the framework of topological data analysis can be extended to general Bregman divergences, widening the scope of possible applications. In particular, we prove that appropriately generalized Cech and Delaunay (alpha) complexes capture the correct homotopy type, namely that of the corresponding union of Bregman balls. Consequently, their filtrations give the correct persistence diagram, namely the one generated by the uniformly growing Bregman balls. Moreover, we show that unlike the metric setting, the filtration of Vietoris-Rips complexes may fail to approximate the persistence diagram. We propose algorithms to compute the thus generalized Cech, Vietoris-Rips and Delaunay complexes and experimentally test their efficiency. Lastly, we explain their surprisingly good performance by making a connection with discrete Morse theory

Are changes in education important for the wage premium and unemployment?

A generalized rise in unemployment rates for both college and high-school graduates, a widening education wage premium, and a sharp increase in college education participation are characteristic features of the transformations of the U.S. labor market between 1970 and 1990. This paper investigates the interactions between these changes in the labor market and in educational attainment. First, it develops an equilibrium search and matching model of the labor market where education is endogenously determined. Second, calibrated versions of the model are used to study quantitatively whether either a skill-biased change in technology or a mismatch shock can explain the above facts. The skill-biased shock accounts for a considerable part of the changes but fails to produce the increase in unemployment for the educated labor force. The mismatch shock explains instead much of the change in the four variables, including the wage premium.Education, wage Premium, unemployment