关于集合上表示函数的限制
On the Limitations of Representing Functions on Sets
摘要：
最近关于集合上函数的表示的工作已经考虑了在潜在空间中使用求和来执行排列不变性。特别是，它已被证实，这潜在的空间的维度可能保持固定的基数的考虑中的集合增加。然而，我们证明，导致这一猜想的分析需要映射是高度不连续的，并认为这只是有限的实际用途。受此观察的启发，我们证明了通过连续映射（例如神经网络或高斯过程）实现该模型实际上对潜在空间的维度施加了约束。对于集合输入的实用泛函数表示只能用至少为输入元素的最大数目的大小的潜在维度来实现。
英文摘要：
Recent work on the representation of functions on sets has considered the use of summation in a latent space to enforce permutation invariance. In particular, it has been conjectured that the dimension of this latent space may remain fixed as the cardinality of the sets under consideration increases. However, we demonstrate that the analysis leading to this conjecture requires mappings which are highly discontinuous and argue that this is only of limited practical use. Motivated by this observation, we prove that an implementation of this model via continuous mappings (as provided by e.g. neural networks or Gaussian processes) actually imposes a constraint on the dimensionality of the latent space. Practical universal function representation for set inputs can only be achieved with a latent dimension at least the size of the maximum number of input elements.
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