  Additive models can be used for interpretable machine learning for their
clarity and simplicity. However, In the classical models for high-order data,
the vectorization operation disrupts the data structure, which may lead to
degenerated accuracy and increased computational complexity. To deal with these
problems, we propose the tensor polynomial addition model (TPAM). It retains
the multidimensional structure information of high-order inputs with tensor
representation. The model parameter compression is achieved using a
hierarchical and low-order symmetric tensor approximation. In this way, complex
high-order feature interactions can be captured with fewer parameters.
Moreover, The TPAM preserves the inherent interpretability of additive models,
facilitating transparent decision-making and the extraction of meaningful
feature values. Additionally, leveraging TPAM's transparency and ability to
handle higher-order features, it is used as a post-processing module for other
interpretation models by introducing two variants for class activation maps.
Experimental results on a series of datasets demonstrate that TPAM can enhance
accuracy by up to 30\%, and compression rate by up to 5 times, while
maintaining a good interpretability.
