# Revisiting Feature Interactions from the Perspective of Quadratic Neural Networks for Click-through Rate Prediction

arXiv: https://arxiv.org/abs/2505.17999 · HTML: https://mlsanity.com/paper/2505.17999/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 78 references that are in the map.
- First posted on arXiv 23 May 2025.

## Builds on (most cited first, 30 of 43)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Averaging Weights Leads to Wider Optima and Better Generalization](https://mlsanity.com/paper/1803.05407/) (2018)
- [LightGCN: Simplifying and Powering Graph Convolution Network for Recommendation](https://mlsanity.com/paper/2002.02126/) (2020)
- [Neural Collaborative Filtering](https://mlsanity.com/paper/1708.05031/) (2017)
- [When Does Label Smoothing Help?](https://mlsanity.com/paper/1906.02629/) (2019)
- [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)
- [DCN V2: Improved Deep & Cross Network and Practical Lessons for Web-scale Learning to Rank Systems](https://mlsanity.com/paper/2008.13535/) (2020)
- [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)
- [R-Drop: Regularized Dropout for Neural Networks](https://mlsanity.com/paper/2106.14448/) (2021)
- [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)
- [Attentional Factorization Machines: Learning the Weight of Feature Interactions via Attention Networks](https://mlsanity.com/paper/1708.04617/) (2017)
- [Neural Collaborative Filtering vs. Matrix Factorization Revisited](https://mlsanity.com/paper/2005.09683/) (2020)
- [PEPNet: Parameter and Embedding Personalized Network for Infusing with Personalized Prior Information](https://mlsanity.com/paper/2302.01115/) (2023)
- [Field-weighted Factorization Machines for Click-Through Rate Prediction in Display Advertising](https://mlsanity.com/paper/1806.03514/) (2018)
- [Fi-GNN: Modeling Feature Interactions via Graph Neural Networks for CTR Prediction](https://mlsanity.com/paper/1910.05552/) (2019)
- [BARS: Towards Open Benchmarking for Recommender Systems](https://mlsanity.com/paper/2205.09626/) (2022)
- [Open Benchmarking for Click-Through Rate Prediction](https://mlsanity.com/paper/2009.05794/) (2020)
- [Adaptive Factorization Network: Learning Adaptive-Order Feature Interactions](https://mlsanity.com/paper/1909.03276/) (2019)
- [MaskNet: Introducing Feature-Wise Multiplication to CTR Ranking Models by Instance-Guided Mask](https://mlsanity.com/paper/2102.07619/) (2021)
- [Product-based Neural Networks for User Response Prediction over Multi-field Categorical Data](https://mlsanity.com/paper/1807.00311/) (2018)
- [Quadratic Residual Networks: A New Class of Neural Networks for Solving Forward and Inverse Problems in Physics Involving PDEs](https://mlsanity.com/paper/2101.08366/) (2021)
- [FinalMLP: An Enhanced Two-Stream MLP Model for CTR Prediction](https://mlsanity.com/paper/2304.00902/) (2023)
- [FM2: Field-matrixed Factorization Machines for Recommender Systems](https://mlsanity.com/paper/2102.12994/) (2021)
- [Tenrec: A Large-scale Multipurpose Benchmark Dataset for Recommender Systems](https://mlsanity.com/paper/2210.10629/) (2022)

## Built on by (most cited first, 3 of 3)

- [FairFS: Addressing Deep Feature Selection Biases for Recommender System](https://mlsanity.com/paper/2602.20001/) (2026)
- [TimeMM: Time-as-Operator Spectral Filtering for Dynamic Multimodal Recommendation](https://mlsanity.com/paper/2604.26247/) (2026)
- [Beyond Interaction Capacity: Estimator Scaling with Recursive Models for CTR Prediction](https://mlsanity.com/paper/2609.37905/) (2026)

## Abstract (arXiv)

Hadamard Product (HP) has long been a cornerstone in click-through rate (CTR) prediction tasks due to its simplicity, effectiveness, and ability to capture feature interactions without additional parameters. However, the underlying reasons for its effectiveness remain unclear. In this paper, we revisit HP from the perspective of Quadratic Neural Networks (QNN), which leverage quadratic interaction terms to model complex feature relationships. We further reveal QNN's ability to expand the feature space and provide smooth nonlinear approximations without relying on activation functions. Meanwhile, we find that traditional post-activation does not further improve the performance of the QNN. Instead, mid-activation is a more suitable alternative. Through theoretical analysis and empirical evaluation of 25 QNN neuron formats, we identify a good-performing variant and make further enhancements on it. Specifically, we propose the Multi-Head Khatri-Rao Product as a superior alternative to HP and a Self-Ensemble Loss with dynamic ensemble capability within the same network to enhance computational efficiency and performance. Ultimately, we propose a novel neuron format, QNN-alpha, which is tailored for CTR prediction tasks. Experimental results show that QNN-alpha achieves new state-of-the-art performance on six public datasets while maintaining low inference latency, good scalability, and excellent compatibility. The code, running logs, and detailed hyperparameter configurations are available at: https://github.com/salmon1802/QNN.