Click-through rate (CTR) prediction plays a critical role in the recommender system and online advertising. The data used in these applications are multi-field categorical data, where each feature belongs to one field. Field information is proved to be important and there are several works considering fields in their models. In this paper, we propose a novel way to model the field information effectively and efficiently, we called it Field-matrixed Factorization Machines (FmFM, or FM^2), which is a direct improvement of FwFMs. We also proposed a new explanation of FMs and FwFMs within the FmFMs framework, and compared the FFMs and FmFMs. Besides pruning the cross terms, our model can support fields specific variable dimensions in embedding, which act as a soft pruning. We also propose an efficient way to minimize the dimension while keeping the model performance. The FmFM model can also be optimized further by cache the intermediate vectors, and it only takes thousands of floating-point operations (FLOP) to make a prediction. Our experiment results show that it can out-perform the FFMs, which is a higher complexity model. The FmFMs model’s performance is also comparable to those complex DNN models which requires millions or more FLOP. We open-sourced our code at https://github.com/fmfm2020/FmFM

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