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It is widely believed that learning good representations is one of the mainreasons for the success of deep neural networks. Although highly intuitive,there is a lack of theory and systematic approach quantitatively characterizingwhat representations do deep neural networks learn. In this work, we move atiny step towards a theory and better understanding of the representations.Specifically, we study a simpler problem: How similar are the representationslearned by two networks with identical architecture but trained from differentinitializations. We develop a rigorous theory based on the neuron activationsubspace match model. The theory gives a complete characterization of thestructure of neuron activation subspace matches, where the core concepts aremaximum match and simple match which describe the overall and the finestsimilarity between sets of neurons in two networks respectively. We alsopropose efficient algorithms to find the maximum match and simple matches.Finally, we conduct extensive experiments using our algorithms. Experimentalresults suggest that, surprisingly, representations learned by the sameconvolutional layers of networks trained from different initializations are notas similar as prevalently expected, at least in terms of subspace match.

Towards Understanding Learning Representations: To What Extent Do Different Neural Networks Learn the Same Representation