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We introduce a new tool for interpreting neural nets, namely full-gradients, which decomposes the neural net response into input sensitivity and per-neuron sensitivity components. This is the first proposed representation which satisfies two key properties: completeness and weak dependence, which provably cannot be satisfied by any saliency map-based interpretability method. Using full-gradients, we also propose an approximate saliency map representation for convolutional nets dubbed FullGrad, obtained by aggregating the full-gradient components.