Hyperplane Arrangements and Fixed Points in Iterated PWL Neural Networks
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Abstract (arXiv)
We leverage the framework of hyperplane arrangements to analyze potential regions of (stable) fixed points. We provide an upper bound on the number of fixed points for multi-layer neural networks equipped with piecewise linear (PWL) activation functions with arbitrary many linear pieces. The theoretical optimality of the exponential growth in the number of layers of the latter bound is shown. Specifically, we also derive a sharper upper bound on the number of stable fixed points for one-hidden-layer networks with hard tanh activation.
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References · 7
- 2014On the Number of Linear Regions of Deep Neural Networkscited by 381
- 2016On the Expressive Power of Deep Neural Networkscited by 306
- 2016Benefits of depth in neural networkscited by 250
- 2017Bounding and Counting Linear Regions of Deep Neural Networkscited by 161
- 2013On the number of response regions of deep feed forward networks with piece-wise linear activationscited by 94
- 2020On the Number of Linear Regions of Convolutional Neural Networkscited by 49
- 2021Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sumscited by 37
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