Fig. 2: Degrees of freedom in data.
From: Inverse imaging with elastic waves driven by unsupervised machine learning
a The structure used to demonstrate inverse imaging: a beam with three blocks on top, with four parameters \({E}_{0},{h}_{1},{h}_{2},\) and \({h}_{3}\) to be recovered. b The standard deviation (Std) of \({\mu }_{i}\) and the mean of \({\sigma }_{i}\) (with \({z}_{i} \sim N({\mu }_{i},{\sigma }_{i}^{2})\) to generate latent variables) across the training dataset, indicating whether a latent variable is meaningful or not. A meaningful variable points to a definite generation with low values of \({\sigma }_{i}\) but with \({\mu }_{i}\) varying with data, in contrast to a meaningless variable, which is random with small \({\mu }_{i}\) and high \({\sigma }_{i}\). We obtain four meaningful latent variables here, exactly corresponding to the number of unknown quantities to be recovered. c The evolutions of the reconstruction error \({L}_{{rec}}\) and the number of meaningful latent variables \({n}_{M}\) as the hyperparameter \(\beta\) varies from \({10}^{-10}\) to \({10}^{1}\), indicating that the number of DOFs in the data is \(n=4\), before \({L}_{{rec}}\) significantly increasing.
