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Metric magnitude of a point cloud is a measure of its ``size. It has been adapted to various mathematical contexts and recent work suggests that it can enhance machine learning and optimization algorithms. But its usability is limited due to the computational cost when the dataset is large or when the computation must be carried out repeatedly (e.g. in model training). In this paper, we study the magnitude computation problem, and show efficient ways of approximating it. We show that it can be cast as a convex optimization problem, but not as a submodular optimization. The paper describes two new algorithms -- an iterative approximation algorithm that converges fast and is accurate in practice, and a subset selection method that makes the computation even faster. It has previously been proposed that the magnitude of model sequences generated during stochastic gradient descent is correlated to the generalization gap. Extension of this result using our more scalable algorithms shows that longer sequences bear higher correlations. We also describe new applications of magnitude in machine learning -- as an effective regularizer for neural network training, and as a novel clustering criterion. 

Abstract Private nonprofit colleges are increasingly using tuition resets, or a decrease in sticker price by at least 5%, to attract new students and counter declining demand. While discounting tuition with institutional aid is a common practice to get accepted students to matriculate and to increase affordability, a tuition reset is a more transparent approach that moves colleges away from a high aid/high tuition model. The authors find minimal evidence that these policies increase student enrollment in the long run, but that there may be short-term impacts. As expected, institutional aid decreases and varies directly with the size of the sticker price reduction. The average net price students pay decreases, but this effect may be driven by changes in the estimated non-tuition elements of the total cost of attendance. Finally, net tuition revenue appears unrelated to tuition resets. These findings call into question the efficacy of this practice.