Geometry and Generalization: Eigenvalues as predictors of where a network will fail to generalize
This provides a dataset-independent method for assessing generalization in autoencoders, which is incremental as it builds on existing geometric analysis techniques.
The paper tackles the problem of predicting where a trained autoencoder will fail to generalize by analyzing the geometry of the input space through Jacobian matrices, showing that the trace and product of eigenvalues are good predictors of mean squared error on test points without needing dataset knowledge.
We study the deformation of the input space by a trained autoencoder via the Jacobians of the trained weight matrices. In doing so, we prove bounds for the mean squared errors for points in the input space, under assumptions regarding the orthogonality of the eigenvectors. We also show that the trace and the product of the eigenvalues of the Jacobian matrices is a good predictor of the MSE on test points. This is a dataset independent means of testing an autoencoder's ability to generalize on new input. Namely, no knowledge of the dataset on which the network was trained is needed, only the parameters of the trained model.