Interaction detection problem is to quantify the influence of any subset of input features that may potentially be an interaction.

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We thank the reviewers for their insightful feedback; we address each review below. R1: "...it is known how to solve the BFE optimisation problem by double loop algorithms" "...what is meant by'they are run once..."' "...meaningless for pairwise marginals..." Agreed. We included the pairwise marginals just for completeness. "Ising model...expect that the estimation quality will degrade with the (average) interaction strength." "The experiments have in my view a preliminary character" We agree our experiments are on small datasets.

Quanshi Zhang is the corresponding author.

Algorithm 1: The proposed Persistence Interaction Detection (PID) algorithmInput: A trained feed-forward neural network, target layer l, norm p. Output: ranked list of interaction candidates {I Our PID framework is presented in Algorithm 1. PID in all experiments of this paper (i.e., set η as 0). In this subsection, we will prove Theorem 1 and evaluate it empirically. We have the following corollary: Corollary 1. |b Combining them together finishes the proof. It is trivial to show that Corollary 1 can be extended to the death time, i.e., we also have After proving Corollary 1, we return to prove the theorem. In this section, first, we show how to extend PID to CNNs.

For introduction to the image task, please see R1: Q2.

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