A Law of Data Reconstruction for Random Features (And Beyond)

Large-scale deep learning models are known to *memorize* parts of the training set. In machine learning theory, memorization is often framed as interpolation or label fitting, and classical results show that this can be achieved when the number of parameters $p$ in the model is larger than the number of training samples $n$. In this work, we consider memorization from the perspective of *data reconstruction*, demonstrating that this can be achieved when $p$ is larger than $dn$, where $d$ is the dimensionality of the data. More specifically, we show that, in the random features model, when $p \gg dn$, the subspace spanned by the training samples in feature space gives sufficient information to identify the individual samples in input space. Our analysis suggests an optimization method to reconstruct the dataset from the model parameters, and we demonstrate that this method performs well on various architectures (random features, two-layer fully-connected and deep residual networks). Our results reveal a *law of data reconstruction*, according to which the entire training dataset can be recovered as $p$ exceeds the threshold $dn$.