Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Expressiveness of Neural Networks Having Width Equal or Below the Input
  Dimension   The understanding about the minimum width of deep neural networks needed to
ensure universal approximation for different activation functions has
progressively been extended (Park et al., 2020). In particular, with respect to
approximation on general compact sets in the input space, a network width less
than or equal to the input dimension excludes universal approximation. In this
work, we focus on network functions of width less than or equal to the latter
critical bound. We prove that in this regime, the exact fit of partially
constant functions on disjoint compact sets is still possible for ReLU network
functions under some conditions on the mutual location of these components. We
show that with cosine as activation function, a three layer network of width
one is sufficient to approximate any function on arbitrary finite sets.
Conversely, we prove a maximum principle from which we conclude that for all
continuous and monotonic activation functions, universal approximation of
arbitrary continuous functions is impossible on sets that coincide with the
boundary of an open set plus an inner point.

27
The categories of this paper are: comp-gas
3
The categories of this paper are: physics
37
The categories of this paper are: supr-con
18
The categories of this paper are: econ
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat
30
The categories of this paper are: q-alg
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The categories of this paper are: plasm-ph
23
The categories of this paper are: solv-int
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The categories of this paper are: nucl-th
34
The categories of this paper are: chem-ph
22
The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-ex
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The categories of this paper are: mtrl-th
36
The categories of this paper are: cmp-lg