The key part is the 'bag of features' discussion. It's SUPER interesting work, demonstrating that classification works just as well if you cut up the image into

The key part is the 'bag of features' discussion. It's SUPER interesting work, demonstrating that classification works just as well if you cut up the image into square chunks and scramble them - the convolutional activations end up roughly similar, but all sense of shape is lost (as is any human recognition of the scrambled objects). I think MNIST is too simple to have much bearing on the problem. MNIST shapes are two-dimensional; actual object classification requires 3D shapes (and all possible projections thereof), which is a much harder problem. Since this is just a message board discussion, my wild conjecture would be that the model has no idea about the existence of the third dimension, or basic physical concepts like lighting and distance. And without that larger world model in which to situate things, texture is maybe the easiest thing to cling to.