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A large number of deep learning architectures use spatial transformations ofCNN feature maps or filters to better deal with variability in objectappearance caused by natural image transformations. In this paper, we provethat spatial transformations of CNN feature maps cannot align the feature mapsof a transformed image to match those of its original, for general affinetransformations, unless the extracted features are themselves invariant. Ourproof is based on elementary analysis for both the single- and multi-layernetwork case. The results imply that methods based on spatial transformationsof CNN feature maps or filters cannot replace image alignment of the input andcannot enable invariant recognition for general affine transformations,specifically not for scaling transformations or shear transformations. Forrotations and reflections, spatially transforming feature maps or filters canenable invariance but only for networks with learnt or hardcoded rotation- orreflection-invariant features

Inability of spatial transformations of CNN feature maps to support invariant recognition