Invariance of reparametrization in model selection of neural networks

The goal of model selection is to identify the model that generated the data. Goodness of a model is measured using generalization, which takes two opposite pressures: goodness of fit and model complexity into account. In the paper we take neural network as an example and use conception of curvature from the point of view of differential geometry to explore the intrinsic model complexity that is free of reparametrization; and then through theoretical analysis, we show the future residual that is qualified to measure the generalization can be expressed by using the intrinsic curvature array of model, from which we give a new model selection criterion, it not only considers the factors such as the number of parameters, sample size and functional form, but also with very clear and intuitive geometric understanding of model selection.