@conference {cBonet22,
	title = {Channel Redundancy and Overlap in Convolutional Neural Networks with Channel-Wise NNK Graphs},
	booktitle = {International Conference on Acoustics, Speech and Signal Processing},
	year = {2022},
	month = {05/2022},
	abstract = {<p>Feature spaces in the deep layers of convolutional neural networks (CNNs) are often very high-dimensional and difficult to interpret. However, convolutional layers consist of multiple channels that are activated by different types of inputs, which suggests that more insights may be gained by studying the channels and how they relate to each other. In this paper, we first analyze theoretically channel-wise non-negative kernel (CW-NNK) regression graphs, which allow us to quantify the overlap between channels and, indirectly, the intrinsic dimension of the data representation manifold. We find that redundancy between channels is significant and varies with the layer depth and the level of regularization during training. Additionally, we observe that there is a correlation between channel overlap in the last convolutional layer and generalization performance. Our experimental results demonstrate that these techniques can lead to a better understanding of deep representations.\&nbsp;</p>},
	author = {Bonet, D. and Ortega, A. and Ruiz-Hidalgo, J. and Shekkizhar, S.}
}
@conference {cBonet21,
	title = {Channel-Wise Early Stopping without a Validation Set via NNK Polytope Interpolation},
	booktitle = {Asia Pacific Signal and Information Processing Association Annual Summit, APSIPA},
	year = {2021},
	month = {12/2021},
	address = {Tokyo, Japan},
	abstract = {<p>State-of-the-art neural network architectures continue to scale in size and deliver impressive generalization results, although this comes at the expense of limited interpretability. In particular, a key challenge is to determine when to stop training the model, as this has a significant impact on generalization. Convolutional neural networks (ConvNets) comprise high-dimensional feature spaces formed by the aggregation of multiple channels, where analyzing intermediate data representations and the model{\textquoteright}s evolution can be challenging owing to the curse of dimensionality. We present channel-wise DeepNNK (CW-DeepNNK), a novel channel-wise generalization estimate based on non-negative kernel regression (NNK) graphs with which we perform local polytope interpolation on low-dimensional channels. This method leads to instance-based interpretability of both the learned data representations and the relationship between channels. Motivated by our observations, we use CW-DeepNNK to propose a novel early stopping criterion that (i) does not require a validation set, (ii) is based on a task performance metric, and (iii) allows stopping to be reached at different points for each channel. Our experiments demonstrate that our proposed method has advantages as compared to the standard criterion based on validation set performance.</p>},
	url = {https://arxiv.org/abs/2107.12972},
	author = {Bonet, D. and Ortega, A. and Ruiz-Hidalgo, J. and Shekkizhar, S.}
}
