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This paper aims to interpret the mechanism of feedforward ReLU networks byexploring their solutions for piecewise linear functions, through the deductionfrom basic rules. The constructed solution should be universal enough toexplain some network architectures of engineering; in order for that, severalways are provided to enhance the solution universality. Some of theconsequences of our theories include: Under affine-geometry background, thesolutions of both three-layer networks and deep-layer networks are given,particularly for those architectures applied in practice, such as multilayerfeedforward neural networks and decoders; We give clear and intuitiveinterpretations of each component of network architectures; Theparameter-sharing mechanism for multi-outputs is investigated; We provide anexplanation of overparameterization solutions in terms of affine transforms;Under our framework, an advantage of deep layers compared to shallower ones isnatural to be obtained. Some intermediate results are the basic knowledge forthe modeling or understanding of neural networks, such as the classification ofdata embedded in a higher-dimensional space, the generalization of affinetransforms, the probabilistic model of matrix ranks, and the concept ofdistinguishable data sets.

Theoretical Exploration of Solutions of Feedforward ReLU Networks