集合上の関数表現の限界について
On the Limitations of Representing Functions on Sets
集合入力の関数を潜在空間の和で表現する手法の理論的限界を示し、連続写像では潜在次元が最大入力要素数以上必要であることを証明した論文。
著者: Edward Wagstaff, Fabian B. Fuchs, Martin Engelcke, Ingmar Posner, Michael Osborne
分類: cs.LG, cs.AI, cs.NE, cs.RO, stat.ML
原文アブストラクト
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.
