# FM2: Field-matrixed Factorization Machines for Recommender Systems

arXiv: https://arxiv.org/abs/2102.12994 · HTML: https://mlsanity.com/paper/2102.12994/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 30 references that are in the map.
- First posted on arXiv 20 Feb 2021.

## Builds on (most cited first, 15 of 15)

- [Wide & Deep Learning for Recommender Systems](https://mlsanity.com/paper/1606.07792/) (2016)
- [DeepFM: A Factorization-Machine based Neural Network for CTR Prediction](https://mlsanity.com/paper/1703.04247/) (2017)
- [Deep Interest Network for Click-Through Rate Prediction](https://mlsanity.com/paper/1706.06978/) (2017)
- [Deep & Cross Network for Ad Click Predictions](https://mlsanity.com/paper/1708.05123/) (2017)
- [AutoInt: Automatic Feature Interaction Learning via Self-Attentive Neural Networks](https://mlsanity.com/paper/1810.11921/) (2018)
- [xDeepFM: Combining Explicit and Implicit Feature Interactions for Recommender Systems](https://mlsanity.com/paper/1803.05170/) (2018)
- [Neural Factorization Machines for Sparse Predictive Analytics](https://mlsanity.com/paper/1708.05027/) (2017)
- [Product-based Neural Networks for User Response Prediction](https://mlsanity.com/paper/1611.00144/) (2016)
- [Deep Learning over Multi-field Categorical Data: A Case Study on User Response Prediction](https://mlsanity.com/paper/1601.02376/) (2016)
- [Field-weighted Factorization Machines for Click-Through Rate Prediction in Display Advertising](https://mlsanity.com/paper/1806.03514/) (2018)
- [Rocket Launching: A Universal and Efficient Framework for Training Well-performing Light Net](https://mlsanity.com/paper/1708.04106/) (2017)
- [Field-aware Factorization Machines in a Real-world Online Advertising System](https://mlsanity.com/paper/1701.04099/) (2017)
- [Predicting Different Types of Conversions with Multi-Task Learning in Online Advertising](https://mlsanity.com/paper/1907.10235/) (2019)
- [OFF-Set: One-pass Factorization of Feature Sets for Online Recommendation in Persistent Cold Start Settings](https://mlsanity.com/paper/1308.1792/) (2013)
- [Pricing Ad Slots with Consecutive Multi-unit Demand](https://mlsanity.com/paper/1308.1382/) (2013)

## Built on by (most cited first, 30 of 31)

- [Click-Through Rate Prediction in Online Advertising: A Literature Review](https://mlsanity.com/paper/2202.10462/) (2022)
- [On the Embedding Collapse when Scaling up Recommendation Models](https://mlsanity.com/paper/2310.04400/) (2023)
- [SAR-Net: A Scenario-Aware Ranking Network for Personalized Fair Recommendation in Hundreds of Travel Scenarios](https://mlsanity.com/paper/2110.06475/) (2021)
- [CausPref: Causal Preference Learning for Out-of-Distribution Recommendation](https://mlsanity.com/paper/2202.03984/) (2022)
- [CL4CTR: A Contrastive Learning Framework for CTR Prediction](https://mlsanity.com/paper/2212.00522/) (2022)
- [A Survey on Fairness-aware Recommender Systems](https://mlsanity.com/paper/2306.00403/) (2023)
- [GraphFM: Graph Factorization Machines for Feature Interaction Modeling](https://mlsanity.com/paper/2105.11866/) (2021)
- [Ads Recommendation in a Collapsed and Entangled World](https://mlsanity.com/paper/2403.00793/) (2024)
- [External Large Foundation Model: How to Efficiently Serve Trillions of Parameters for Online Ads Recommendation](https://mlsanity.com/paper/2502.17494/) (2025)
- [Memorize, Factorize, or be Na"ive: Learning Optimal Feature Interaction Methods for CTR Prediction](https://mlsanity.com/paper/2108.01265/) (2021)
- [Hierarchical LoRA MoE for Efficient CTR Model Scaling](https://mlsanity.com/paper/2510.10432/) (2025)
- [Temporal Interest Network for User Response Prediction](https://mlsanity.com/paper/2308.08487/) (2023)
- [InterFormer: Effective Heterogeneous Interaction Learning for Click-Through Rate Prediction](https://mlsanity.com/paper/2411.09852/) (2024)
- [MerRec: A Large-scale Multipurpose Mercari Dataset for Consumer-to-Consumer Recommendation Systems](https://mlsanity.com/paper/2402.14230/) (2024)
- [Field Matters: A Lightweight LLM-enhanced Method for CTR Prediction](https://mlsanity.com/paper/2505.14057/) (2025)
- [Boosting Factorization Machines via Saliency-Guided Mixup](https://mlsanity.com/paper/2206.08661/) (2022)
- [Hierarchically Fusing Long and Short-Term User Interests for Click-Through Rate Prediction in Product Search](https://mlsanity.com/paper/2304.02089/) (2023)
- [Robust Uplift Modeling with Large-Scale Contexts for Real-time Marketing](https://mlsanity.com/paper/2502.15697/) (2025)
- [Revisiting Feature Interactions from the Perspective of Quadratic Neural Networks for Click-through Rate Prediction](https://mlsanity.com/paper/2505.17999/) (2025)
- [TokenFormer: Unify the Multi-Field and Sequential Recommendation Worlds](https://mlsanity.com/paper/2604.13737/) (2026)
- [AutoAttention: Automatic Field Pair Selection for Attention in User Behavior Modeling](https://mlsanity.com/paper/2210.15154/) (2022)
- [Low Rank Field-Weighted Factorization Machines for Low Latency Item Recommendation](https://mlsanity.com/paper/2408.00801/) (2024)
- [PHN: Parallel heterogeneous network with soft gating for CTR prediction](https://mlsanity.com/paper/2206.09184/) (2022)
- [Directed Acyclic Graph Factorization Machines for CTR Prediction via Knowledge Distillation](https://mlsanity.com/paper/2211.11159/) (2022)
- [AdaF^2M^2: Comprehensive Learning and Responsive Leveraging Features in Recommendation System](https://mlsanity.com/paper/2501.15816/) (2025)
- [From Data to Decisions: The Transformational Power of Machine Learning in Business Recommendations](https://mlsanity.com/paper/2402.08109/) (2024)
- [Large Foundation Model for Ads Recommendation](https://mlsanity.com/paper/2508.14948/) (2025)
- [Toward a benchmark for CTR prediction in online advertising: datasets, evaluation protocols and perspectives](https://mlsanity.com/paper/2512.01179/) (2025)
- [tensorFM: Low-Rank Approximations of Cross-Order Feature Interactions](https://mlsanity.com/paper/2602.15229/) (2026)
- [Understanding DNNs in Feature Interaction Models: A Dimensional Collapse Perspective](https://mlsanity.com/paper/2604.26489/) (2026)

## Abstract (arXiv)

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 FM2). 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.