Click-through rate (CTR) prediction plays a critical role in recommender
systems 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 proposed a novel approach to model
the field information effectively and efficiently. The proposed approach is a
direct improvement of FwFM, and is named as Field-matrixed Factorization
Machines (FmFM, or $FM^2$). We also proposed a new explanation of FM and FwFM
within the FmFM framework, and compared it with the FFM. Besides pruning the
cross terms, our model supports field-specific variable dimensions of embedding
vectors, which acts as soft pruning. We also proposed an efficient way to
minimize the dimension while keeping the model performance. The FmFM model can
also be optimized further by caching the intermediate vectors, and it only
takes thousands of floating-point operations (FLOPs) to make a prediction. Our
experiment results show that it can out-perform the FFM, which is more complex.
The FmFM model's performance is also comparable to DNN models which require
much more FLOPs in runtime.

该研究提出了一种新的方法，名为 Field-matrixed Factorization Machines（FmFM，或 FM^2），用于有效和高效地建模数据中的字段信息，在交叉项剪枝的基础上支持字段特定变量维度的嵌入向量，并通过缓存中间向量进一步优化模型，实验结果显示该方法可以优于 FFM 和 DNN 模型。