  The training dynamics of hidden layers in deep learning are poorly understood
in theory. Recently, the Information Plane (IP) was proposed to analyze them,
which is based on the information-theoretic concept of mutual information (MI).
The Information Bottleneck (IB) theory predicts that layers maximize relevant
information and compress irrelevant information. Due to the limitations in MI
estimation from samples, there is an ongoing debate about the properties of the
IP for the supervised learning case. In this work, we derive a theoretical
convergence for the IP of autoencoders. The theory predicts that ideal
autoencoders with a large bottleneck layer size do not compress input
information, whereas a small size causes compression only in the encoder
layers. For the experiments, we use a Gram-matrix based MI estimator recently
proposed in the literature. We propose a new rule to adjust its parameters that
compensates scale and dimensionality effects. Using our proposed rule, we
obtain experimental IPs closer to the theory. Our theoretical IP for
autoencoders could be used as a benchmark to validate new methods to estimate
MI in neural networks. In this way, experimental limitations could be
recognized and corrected, helping with the ongoing debate on the supervised
learning case.
