Recently, there has been significant attention on determining the minimum width for the universal approximation property of deep, narrow MLPs. Among these challenges, approximating a continuous function under the uniform norm is important and challenging, with the gap between its lower and upper bound being hard to narrow. In this regard, we propose a novel upper bound for the minimum width, given by $\operatorname{max}(2d_x+1, d_y) + \alpha(\sigma)$, to achieve uniform approximation in deep narrow MLPs, where $0\leq \alpha(\sigma)\leq 2$ represents the constant depending on the activation function. We demonstrate this bound through two key proofs. First, we establish that deep, narrow MLPs with little additional width can approximate diffeomorphisms. Secondly, we utilize the Whitney embedding theorem to show that any continuous function can be approximated by embeddings, further decomposed into linear transformations and diffeomorphisms.

翻译：近年来，关于深度窄MLP通用逼近性质所需最小宽度的研究备受关注。其中，一致范数下连续函数逼近问题尤为重要且具有挑战性，其下界与上界之间的差距难以缩小。针对这一难题，我们提出了最小宽度的新上界：$\operatorname{max}(2d_x+1, d_y) + \alpha(\sigma)$，该上界可实现深度窄MLP的一致逼近，其中$0\leq \alpha(\sigma)\leq 2$为依赖于激活函数的常数。我们通过两个关键证明来论证该上界：首先证明仅需少量额外宽度，深度窄MLP即可逼近微分同胚；其次利用Whitney嵌入定理说明任意连续函数均可通过嵌入函数逼近，并可进一步分解为线性变换与微分同胚的组合。