local inverse of a neural network

Question

I have a neural network with N input nodes and N output nodes, and possibly multiple hidden layers and recurrences in it but let's forget about those first. The goal of the neural network is to learn an N-dimensional variable Y*, given N-dimensional value X. Let's say the output of the neural network is Y, which should be close to Y* after learning. My question is: is it possible to get the inverse of the neural network for the output Y*? That is, how do I get the value X* that would yield Y* when put in the neural network? (or something close to it)

A major part of the problem is that N is very large, typically in the order of 10000 or 100000, but if anyone knows how to solve this for small networks with no recurrences or hidden layers that might already be helpful. Thank you.

Answer 1

If you can choose the neural network such that the number of nodes in each layer is the same, and the weight matrix is non-singular, and the transfer function is invertible (e.g. leaky relu), then the function will be invertible.

This kind of neural network is simply a composition of matrix multiplication, addition of bias and transfer function. To invert, you'll just need to apply the inverse of each operation in the reverse order. I.e. take the output, apply the inverse transfer function, multiply it by the inverse of the last weight matrix, minus the bias, apply the inverse transfer function, multiply it by the inverse of the second to last weight matrix, and so on and so forth.