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Owen and Hoyt recently showed that the effective dimension offers keystructural information about the input-output mapping underlying an artificialneural network. Along this line of research, this work proposes an estimationprocedure that allows the calculation of the mean dimension from a givendataset, without resampling from external distributions. The design yieldstotal indices when features are independent and a variant of total indices whenfeatures are correlated. We show that this variant possesses the zeroindependence property. With synthetic datasets, we analyse how the meandimension evolves layer by layer and how the activation function impacts themagnitude of interactions. We then use the mean dimension to study some of themost widely employed convolutional architectures for image recognition (LeNet,ResNet, DenseNet). To account for pixel correlations, we propose calculatingthe mean dimension after the addition of an inverse PCA layer that allows oneto work on uncorrelated PCA-transformed features, without the need to retrainthe neural network. We use the generalized total indices to produce heatmapsfor post-hoc explanations, and we employ the mean dimension on thePCA-transformed features for cross comparisons of the artificial neuralnetworks structures. Results provide several insights on the difference inmagnitude of interactions across the architectures, as well as indications onhow the mean dimension evolves during training.

The Mean Dimension of Neural Networks -- What causes the interaction effects?