宽度有界神经网络的吸引盆拓扑性质与表达能力
Topological properties of basins of attraction and expressiveness of width bounded neural networks
AI总结：
本文研究宽度不超过输入维度的神经网络，证明其吸引盆有界且补集无有界连通分量，并在更严格条件下证明吸引盆路径连通，同时解释了此类标量值网络在连续函数空间中不稠密的根本原因。
摘要：
在 Radhakrishnan 等人 [2020] 的工作中，作者通过实验表明，使用常规 SGD 方法训练的自编码器会在其训练数据周围形成吸引盆。我们考虑宽度不超过输入维度的网络函数，并证明在这种情况下吸引盆是有界的，且其补集不可能具有有界连通分量。我们的这些结果所依赖的条件在后者的若干实验中得到满足，因此我们回答了其中提出的一个问题。我们还证明，在一些更严格的条件下，吸引盆是路径连通的。我们通过若干例子证明了结果中条件的紧密性。最后，用于证明上述结果的论证使我们能够推导出一个根本原因，说明满足我们宽度有界条件的标量值神经网络函数为何在连续函数空间中不稠密。
英文摘要：
In Radhakrishnan et al. [2020], the authors empirically show that autoencoders trained with usual SGD methods shape out basins of attraction around their training data. We consider network functions of width not exceeding the input dimension and prove that in this situation basins of attraction are bounded and their complement cannot have bounded components. Our conditions in these results are met in several experiments of the latter work and we thus address a question posed therein. We also show that under some more restrictive conditions the basins of attraction are path-connected. The tightness of the conditions in our results is demonstrated by means of several examples. Finally, the arguments used to prove the above results allow us to derive a root cause why scalar-valued neural network functions that fulfill our bounded width condition are not dense in spaces of continuous functions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
