# How to build in symmetry of inputs into a Deep Neural Network?

I have a Deep Neural Network that takes $$n$$ inputs $$X = [X_1, \ldots, X_n]^T$$ and gives $$n$$ ouputs $$Y = [Y_1, \ldots, Y_n]^T$$. Normally, I can just do a standard deep neural network with a few fully connected hidden layers. However, I want to build into the network the fact that there should be permutational symmetry. That is to say, if $$\pi(\cdot)$$ is a particular permutation, then $$\pi(X)$$ should output $$\pi(Y)$$. So each input should be treated symmetrically in some sense.

A simple example would be I get $$n$$ IID sensor readings, and I want to build an auto-encoder for these sensor readings.

Is there some way of building this symmetry into the neural network, perhaps through some parameter sharing or a special architecture?

• What you're talking about here is the neural net being equivariant under the permutation action, so maybe you could try adapting some techniques from group equivariant convolutional neural networks. – Alexander Gruber Nov 21 '19 at 3:24