Reconstructing and Analyzing Effective Hypersurfaces From Convolutional Neural Network Layers Using AdjointBackMap

Abstract

There are several methods in the exploration of Convolutional Neural Network’s (CNN’s) inner workings. However, in general, finding the inverse of the function performed by CNN as a whole is an ill-posed problem. We propose an Adjoint Operator-based method to reconstruct, given an arbitrary unit in the CNN (except for the first convolutional layer), its effective hypersurface in the input space that replicates the unit’s decision surface conditioned on a particular input image. We gradually study CNN’s inner workings through two steps. First, we consider a CNN without any bias for the reconstruction, which reduces the difficulties in the analysis. Next, we embed input images into an enlarged space (that considers bias as a part of the input) to enable the reconstruction of CNN’s processing that includes bias vectors. Both steps confirm that any reconstructed effective hypersurface would give nearly the exact output value of that CNN unit when an inner product is computed with the original input. Also, we find that CNN unit’s decision is primarily conditioned on the input. Further analysis in adversarial attacks reveals that CNN’s decision is very sensitive and brittle, explaining why adversarial examples can effectively deceive CNNs.