Assessing Neural Network Representations During Training Using
  Noise-Resilient Diffusion Spectral Entropy
Danqi Liao Chen Liu Benjamin W. Christensen Alexander Tong Guillaume Huguet Guy Wolf Maximilian Nickel Ian Adelstein Smita Krishnaswamy
Entropy and mutual information in neural networks provide rich information on
the learning process, but they have proven difficult to compute reliably in
high dimensions. Indeed, in noisy and high-dimensional data, traditional
estimates in ambient dimensions approach a fixed entropy and are prohibitively
hard to compute. To address these issues, we leverage data geometry to access
the underlying manifold and reliably compute these information-theoretic
measures. Specifically, we define diffusion spectral entropy (DSE) in neural
representations of a dataset as well as diffusion spectral mutual information
(DSMI) between different variables representing data. First, we show that they
form noise-resistant measures of intrinsic dimensionality and relationship
strength in high-dimensional simulated data that outperform classic Shannon
entropy, nonparametric estimation, and mutual information neural estimation
(MINE). We then study the evolution of representations in classification
networks with supervised learning, self-supervision, or overfitting. We observe
that (1) DSE of neural representations increases during training; (2) DSMI with
the class label increases during generalizable learning but stays stagnant
during overfitting; (3) DSMI with the input signal shows differing trends: on
MNIST it increases, while on CIFAR-10 and STL-10 it decreases. Finally, we show
that DSE can be used to guide better network initialization and that DSMI can
be used to predict downstream classification accuracy across 962 models on
ImageNet. The official implementation is available at
https://github.com/ChenLiu-1996/DiffusionSpectralEntropy.