The polytopal complex as a framework to analyze multilayer relu networks
Keywords: theory of deep learning + mlp + low dimension + polytopal complex
TL;DR: We capture the local properties of a mlp such as continuity, number of cells, and extrema by computing its polytopal cell complex.
Abstract: Neural networks have shown superior performance in many different domains.
However, a precise understanding of what even simple architectures actually are
doing is not yet achieved, hindering the application of such architectures in safety critical
embedded systems. To improve this understanding, we think of a network
as a continuous piecewise linear function. The network decomposes the input space
into cells in which the network is an affine function; the resulting cells form a
polytopal complex. In this paper we provide an algorithm to derive this complex.
Furthermore, we capture the local and global behavior of the network by computing
the maxima, minima, number of cells, local span, and curvature of the complex.
With the machinery presented in this paper we can extend the validity of a neural
network beyond the finite discrete test set to an open neighborhood of this test set,
potentially covering large parts of the input domain. To show the effectiveness of
the proposed method we run various experiments on the effects of width, depth,
regularisation, and initial seed on these measures. We empirically confirm that
the solution found by training is strongly influenced by weight initialization. We
further find that under regularization, less cells capture more of the volume, while
the total number of cells stays in the same range. At the same time the total number
of cells stays in the same range. Together, these findings provide novel insights
into the network and its training parameters.
Primary Area: interpretability and explainable AI
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Submission Number: 10614
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