How to understand GArray and GFuncArray?

[shuida]

Hello Dr.Cohen, this is another question. I didn't catch the meaning of GArray and GFuncArray: A GFuncArray is an array of functions on a group G. A GArray represents an array that contains transformations instead of scalars. Can you explain them more clearly?

[tscohen]

Numpy has the ndarray class, which has some great features like reshaping, broadcasting, etc. The idea is to have an array that has the same features, but stores not numbers but group elements or functions (feature maps). Using these abstractions we can implement the GConv in a way that exactly mirrors the mathematics in the paper. Specifically, if you look at make_gconv_indices.py, you can see that for each group, we just have to create a GFuncArray for the filter (we think of the filter as a function on G) and then apply each element of a group to it to create the transformed filters (in one single expression). E.g. for C4 we do x = np.random.randn(4, ksize, ksize); f = P4FuncArray(v=x); i = f.left_translation_indices(C4[:, None, None, None]).

In hindsight however, I think it would have been more clear to just create the index arrays directly without using any abstractions. If you feel like it, you could look at the output of the functions in make_gconv_indices.py (e.g. make_c4_z2_indices()), and write a simple loop to output the same indexing arrays. All the indexing arrays do in the end, is rotate the kernels and permute the orientation channels (if any). The only thing that happens during inference is basically conv2d(feature_maps, weights[precomputed_indices]). A simple / minimal implementation would be a great learning resource for people new to group convolutions. If you keep the code super simple and test it well, you could even try to get it merged into pytorch or tensorflow.

[shuida]

Got it. Thank you so much!

[mvoelk]

I wrote a super simple Keras layer. https://github.com/mvoelk/keras_layers

[tscohen]

Hi Markus, this looks great, thanks. I've been thinking for a long time that it would be nice if someone did a less abstract implementation...

Just a minor comment about the statement "The computation performed in the layer is still slower compared to standard convolution." -- This is true if one uses more channels (they get multiplied by 4 or 8), but if one uses the same number of channels then the difference is negligible. The only overhead is filter expansion. At test time, one can convert the G-CNN to a CNN by doing the filter expansion once, and then there is no overhead. Would be nice if you could mention this in the docs.

Another thing I've found to be useful is to add unit tests that check for equivariance.

[mvoelk]

I did experiments on CIFAR-10 with Conv2D 96 features and GroupConv2D 12 * 8 features. Conv2D takes 22ms/step, GroupConv2D 34ms/step. I assume that it is a matter of implementation. I do more reshapes on the kernel and also reshapes on the features, but that should not be so expensive... In this case I do not think that we get faster due to the fact that we have fewer parameters. What are your experiences?

channels (they get multiplied by 4 or 8)

I should mention that somewhere.

At test time, one can convert the G-CNN to a CNN by doing the filter expansion once, and then there is no overhead.

Good idea, I haven't thought of it yet.

The example in examples.ipynb checks for equivariance.