Integrating geometries of ReLU feedforward neural networks

Abstract

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This paper investigates the integration of multiple geometries present within a ReLU-based neural network. A ReLU neural network determines a piecewise affine linear continuous map, M, from an input space ℝm to an output space ℝn. The piecewise behavior corresponds to a polyhedral decomposition of ℝm. Each polyhedron in the decomposition can be labeled with a binary vector (whose length equals the number of ReLU nodes in the network) and with an affine linear function (which agrees with M when restricted to points in the polyhedron). We develop a toolbox that calculates the binary vector for a polyhedra containing a given data point with respect to a given ReLU FFNN. We utilize this binary vector to derive bounding facets for the corresponding polyhedron, extraction of “active” bits within the binary vector, enumeration of neighboring binary vectors, and visualization of the polyhedral decomposition (Python code is available at https://github.com/cglrtrgy/GoL_Toolbox). Polyhedra in the polyhedral decomposition of ℝm are neighbors if they share a facet. Binary vectors for neighboring polyhedra differ in exactly 1 bit. Using the toolbox, we analyze the Hamming distance between the binary vectors for polyhedra containing points from adversarial/nonadversarial datasets revealing distinct geometric properties. A bisection method is employed to identify sample points with a Hamming distance of 1 along the shortest Euclidean distance path, facilitating the analysis of local geometric interplay between Euclidean geometry and the polyhedral decomposition along the path. Additionally, we study the distribution of Chebyshev centers and related radii across different polyhedra, shedding light on the polyhedral shape, size, clustering, and aiding in the understanding of decision boundaries.

Keywords

• ReLU feedforward neural networks
• binary vectors
• polyhedral decomposition
• geometries
• Chebyshev center
• Hamming distance