Why do CNNs Learn Consistent Representations in their First Layer Independent of Labels and Architecture?

It has previously been observed that the filters learned in the first layer of a CNN are qualitatively similar for different networks and tasks. We extend this finding and show a high quantitative similarity between filters learned by different networks. We consider the CNN filters as a filter bank and measure the sensitivity of the filter bank to different frequencies. We show that the sensitivity profile of different networks is almost identical, yet far from initialization. Remarkably, we show that it remains the same even when the network is trained with random labels. To understand this effect, we derive an analytic formula for the sensitivity of the filters in the first layer of a linear CNN. We prove that when the average patch in images of the two classes is identical, the sensitivity profile of the filters in the first layer will be identical in expectation when using the true labels or random labels and will only depend on the second-order statistics of image patches. We empirically demonstrate that the average patch assumption holds for realistic datasets. Finally we show that the energy profile of filters in nonlinear CNNs is highly correlated with the energy profile of linear CNNs and that our analysis of linear networks allows us to predict when representations learned by state-of-the-art networks trained on benchmark classification tasks will depend on the labels.