I don’t think “sparse neural networks” fit the bill. All the references I’ve turned up for the phrase talk about the usual sort of what I’ve been calling layered NNs, but where most of the parameters are zero. This leaves intact the layer structure.
To express more precisely the sort of connectivity I’m talking about, for any NN, construct the following directed graph. There is one node for every neuron, and an arc from each neuron A to each neuron B whose output depends directly on an output value of A.
For the NNs as described in e.g. Andrej Karpathy’s lectures (which I’m currently going through), this graph is a DAG. Furthermore, it is a DAG having the property of layeredness, which I define thus:
A DAG is layered if every node A can be assigned an integer label L(A), such that for every edge from A to B, L(B) = L(A)+1. A layer is the set of all the nodes having a given label.
The sparse NNs I’ve found in the literature are all layered. A “full” (i.e. not sparse) NN would also satisfy the converse of the above definition, i.e. L(B) = L(A)+1 would imply an edge from A to B.
The simplest example of a non-layered DAG is one with three nodes A, B, and C, with edges from A to B, A to C, and B to C. If you tried to structure this into layers, you would either find an edge between two nodes in the same layer, or an edge that skips a layer.
To cover non-DAG NNs also, I’d call one layered if in the above definition, “L(B) = L(A)+1” is replaced by “L(B) = L(A) ± 1″. (ETA: This is equivalent to the graph being bipartite: the nodes can be divided into two sets such that every edge goes from a node in one set to a node in the other.)
It could be called approximately layered if most edges satisfy the condition.
Are there any not-even-approximately-layered NNs in the literature?
I’m fairly sure that there’s architectures where each layer is a linear function of the concatenated activations of all previous layers, though I can’t seem to find it right now. If you add possible sparsity to that, then I think you get a fully general DAG.
I think you might want to look at the litterature on “sparse neural networks”, which is the right search term for what you mean here.
I don’t think “sparse neural networks” fit the bill. All the references I’ve turned up for the phrase talk about the usual sort of what I’ve been calling layered NNs, but where most of the parameters are zero. This leaves intact the layer structure.
To express more precisely the sort of connectivity I’m talking about, for any NN, construct the following directed graph. There is one node for every neuron, and an arc from each neuron A to each neuron B whose output depends directly on an output value of A.
For the NNs as described in e.g. Andrej Karpathy’s lectures (which I’m currently going through), this graph is a DAG. Furthermore, it is a DAG having the property of layeredness, which I define thus:
A DAG is layered if every node A can be assigned an integer label L(A), such that for every edge from A to B, L(B) = L(A)+1. A layer is the set of all the nodes having a given label.
The sparse NNs I’ve found in the literature are all layered. A “full” (i.e. not sparse) NN would also satisfy the converse of the above definition, i.e. L(B) = L(A)+1 would imply an edge from A to B.
The simplest example of a non-layered DAG is one with three nodes A, B, and C, with edges from A to B, A to C, and B to C. If you tried to structure this into layers, you would either find an edge between two nodes in the same layer, or an edge that skips a layer.
To cover non-DAG NNs also, I’d call one layered if in the above definition, “L(B) = L(A)+1” is replaced by “L(B) = L(A) ± 1″. (ETA: This is equivalent to the graph being bipartite: the nodes can be divided into two sets such that every edge goes from a node in one set to a node in the other.)
It could be called approximately layered if most edges satisfy the condition.
Are there any not-even-approximately-layered NNs in the literature?
I’m fairly sure that there’s architectures where each layer is a linear function of the concatenated activations of all previous layers, though I can’t seem to find it right now. If you add possible sparsity to that, then I think you get a fully general DAG.