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Finding a low-dimensional embedding of a graph ofnnodes inis an essential task in many applications. For instance, maximum variance unfolding (MVU) is a well-known dimensionality reduction method that involves solving this problem. The standard approach is to formulate the embedding problem as a semidefinite program (SDP). However, the SDP approach does not scale well to large graphs. In this paper, we exploit the fact that many graphs have an intrinsically low dimension, and thus the optimal matrix resulting from the solution of the SDP has a low rank. This observation leads to a quadratic reformulation of the SDP that has far fewer variables, but on the other hand, is a difficult convex maximization problem. We propose an approach for obtaining a solution to the SDP by solving a sequence of smaller quadratic problems with increasing dimension. Utilizing an augmented Lagrangian and an interior-point method for solving the quadratic problems, we demonstrate with numerical experiments on MVU problems that the proposed approach scales well to very large graphs. [ABSTRACT FROM AUTHOR]

The Finnish educational system is known for its equality. However, in many key areas in national and international assessments, Swedish-language schools in Finland have lagged behind their Finnish-language counterparts. So far there is little research into the underlying reasons for this discrepancy. In this article, in order to illuminate the phenomenon, we take a closer look at a number of school-level indices based on the Finnish OECD Programme for International Student Assessment (PISA) 2009 data and compare school principals’ responses across these two school categories. Our findings reveal differences in school policies and practices, in school climate, and in school resources between the two parallel language-specific sectors of basic education in Finland. [ABSTRACT FROM PUBLISHER]