论文标题
吸引力盆地的拓扑特性和宽度有界神经网络的表现力
Topological properties of basins of attraction and expressiveness of width bounded neural networks
论文作者
论文摘要
在Radhakrishnan等人中。 [2020]，作者从经验上表明，经常使用SGD方法训练的自动编码器围绕其培训数据塑造了吸引力的盆地。我们认为宽度的网络函数不超过输入维度，并证明在这种情况下，吸引力的盆地是有限的，它们的补体不能具有有限的组件。在后一种工作的几个实验中，我们在这些结果中满足了我们的条件，因此我们解决了其中提出的问题。我们还表明，在一些更限制的条件下，吸引的盆地是路径连接的。通过几个示例证明了我们结果中条件的紧密度。最后，用于证明上述结果的参数使我们能够得出一个根本原因，为什么在连续功能的空间中，满足我们有限宽度条件的标量值神经网络函数并不密集。
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.
