Minimum Width of Leaky-ReLU Neural Networks for Uniform Universal Approximation
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The study of universal approximation properties (UAP) for neural networks (NN) has a long history. When the network width is unlimited, only a single hidden layer is sufficient for UAP. In contrast, when the depth is unlimited, the width for UAP needs to be not less than the critical width w^*_min=max(d_x,d_y), where d_x and d_y are the dimensions of the input and output, respectively. Recently, <cit.> shows that a leaky-ReLU NN with this critical width can achieve UAP for L^p functions on a compact domain K, i.e., the UAP for L^p(K,ℝ^d_y). This paper examines a uniform UAP for the function class C(K,ℝ^d_y) and gives the exact minimum width of the leaky-ReLU NN as w_min=max(d_x+1,d_y)+1_d_y=d_x+1, which involves the effects of the output dimensions. To obtain this result, we propose a novel lift-flow-discretization approach that shows that the uniform UAP has a deep connection with topological theory.
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