Adjoint operators have been found to be effective in the exploration of CNN's inner workings [1]. However, the previous no-bias assumption restricted its generalization. We overcome the restriction via embedding input images into an extended normed space that includes bias in all CNN layers as part of the extended space and propose an adjoint-operator-based algorithm that maps high-level weights back to the extended input space for reconstructing an effective hypersurface. Such hypersurface can be computed for an arbitrary unit in the CNN, and we prove that this reconstructed hypersurface, when multiplied by the original input (through an inner product), will precisely replicate the output value of each unit. We show experimental results based on the CIFAR-10 and CIFAR-100 data sets where the proposed approach achieves near 0 activation value reconstruction error.

翻译：已有研究表明，伴随算子在探索卷积神经网络内部机制方面具有有效性[1]，但先前基于无偏置假设的方法限制了其泛化能力。为突破这一局限，本文通过将输入图像嵌入到包含所有卷积层偏置的扩展赋范空间中，提出了一种基于伴随算子的算法，可将高层权重映射回扩展输入空间以重构有效超曲面。该超曲面可针对CNN中的任意单元进行计算，且我们证明：当通过内积与原输入相乘时，重构的超曲面能精确复现每个单元的输出值。基于CIFAR-10和CIFAR-100数据集的实验表明，本方法实现了接近零的激活值重构误差。