%0 Journal Article %1 wagstaff2019limitations %A Wagstaff, Edward %A Fuchs, Fabian B. %A Engelcke, Martin %A Posner, Ingmar %A Osborne, Michael %D 2019 %K invariance probability readings symmetry theory %T On the Limitations of Representing Functions on Sets %U http://arxiv.org/abs/1901.09006 %X 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.

@article{wagstaff2019limitations, abstract = {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.}, added-at = {2019-11-08T02:03:05.000+0100}, author = {Wagstaff, Edward and Fuchs, Fabian B. and Engelcke, Martin and Posner, Ingmar and Osborne, Michael}, biburl = {https://www.bibsonomy.org/bibtex/20e0536438563c7577fb51b3f452826a7/kirk86}, description = {[1901.09006] On the Limitations of Representing Functions on Sets}, interhash = {c2a20a833cee0bf4cbfbb7c30afc44ad}, intrahash = {0e0536438563c7577fb51b3f452826a7}, keywords = {invariance probability readings symmetry theory}, note = {cite arxiv:1901.09006Comment: Published at the International Conference on Machine Learning (2019)}, timestamp = {2019-11-08T02:03:05.000+0100}, title = {On the Limitations of Representing Functions on Sets}, url = {http://arxiv.org/abs/1901.09006}, year = 2019 }