Generic regularity of conservative solutions to Camassa-Holm type equations

This paper mainly proves the generic properties of the Camassa-Holm equation and the two-component Camassa-Holm equation by Thom's transversality Lemma. We reveal their differences in generic regularity and singular behavior

Technical Note: Defining and Quantifying AND-OR Interactions for Faithful and Concise Explanation of DNNs

In this technical note, we aim to explain a deep neural network (DNN) by quantifying the encoded interactions between input variables, which reflects the DNN's inference logic. Specifically, we first rethink the definition of interactions, and then formally define faithfulness and conciseness for interaction-based explanation. To this end, we propose two kinds of interactions, i.e., the AND interaction and the OR interaction. For faithfulness, we prove the uniqueness of the AND (OR) interaction in quantifying the effect of the AND (OR) relationship between input variables. Besides, based on AND-OR interactions, we design techniques to boost the conciseness of the explanation, while not hurting the faithfulness. In this way, the inference logic of a DNN can be faithfully and concisely explained by a set of symbolic concepts.Comment: arXiv admin note: text overlap with arXiv:2111.0620