# EulerNet: Adaptive Feature Interaction Learning via Euler's Formula for CTR Prediction

arXiv: https://arxiv.org/abs/2304.10711 · HTML: https://mlsanity.com/paper/2304.10711/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Apr 2023.

## Built on by (most cited first, 26 of 26)

- [On the Embedding Collapse when Scaling up Recommendation Models](https://mlsanity.com/paper/2310.04400/) (2023)
- [Ads Recommendation in a Collapsed and Entangled World](https://mlsanity.com/paper/2403.00793/) (2024)
- [The Efficiency vs. Accuracy Trade-off: Optimizing RAG-Enhanced LLM Recommender Systems Using Multi-Head Early Exit](https://mlsanity.com/paper/2501.02173/) (2025)
- [MerRec: A Large-scale Multipurpose Mercari Dataset for Consumer-to-Consumer Recommendation Systems](https://mlsanity.com/paper/2402.14230/) (2024)
- [EulerFormer: Sequential User Behavior Modeling with Complex Vector Attention](https://mlsanity.com/paper/2403.17729/) (2024)
- [Fusion Matters: Learning Fusion in Deep Click-through Rate Prediction Models](https://mlsanity.com/paper/2411.15731/) (2024)
- [Revisiting Feature Interactions from the Perspective of Quadratic Neural Networks for Click-through Rate Prediction](https://mlsanity.com/paper/2505.17999/) (2025)
- [Graph-enhanced Optimizers for Structure-aware Recommendation Embedding Evolution](https://mlsanity.com/paper/2310.03032/) (2023)
- [Beyond KAN: Introducing KarSein for Adaptive High-Order Feature Interaction Modeling in CTR Prediction](https://mlsanity.com/paper/2408.08713/) (2024)
- [AdaF^2M^2: Comprehensive Learning and Responsive Leveraging Features in Recommendation System](https://mlsanity.com/paper/2501.15816/) (2025)
- [Scaled Supervision is an Implicit Lipschitz Regularizer](https://mlsanity.com/paper/2503.14813/) (2025)
- [Heterogeneous Influence Maximization in User Recommendation](https://mlsanity.com/paper/2508.13517/) (2025)
- [FedUD: Exploiting Unaligned Data for Cross-Platform Federated Click-Through Rate Prediction](https://mlsanity.com/paper/2407.18472/) (2024)
- [NeSHFS: Neighborhood Search with Heuristic-based Feature Selection for Click-Through Rate Prediction](https://mlsanity.com/paper/2409.08703/) (2024)
- [Pilot-guided Multimodal Semantic Communication for Audio-Visual Event Localization](https://mlsanity.com/paper/2412.06208/) (2024)
- [Irrational Complex Rotations Empower Low-bit Optimizers](https://mlsanity.com/paper/2501.12896/) (2025)
- [Intrinsic and Extrinsic Factor Disentanglement for Recommendation in Various Context Scenarios](https://mlsanity.com/paper/2503.03524/) (2025)
- [Enhancing Recommender Systems Using Textual Embeddings from Pre-trained Language Models](https://mlsanity.com/paper/2504.08746/) (2025)
- [Open-Set Living Need Prediction with Large Language Models](https://mlsanity.com/paper/2506.02713/) (2025)
- [Lighter-X: An Efficient and Plug-and-play Strategy for Graph-based Recommendation through Decoupled Propagation](https://mlsanity.com/paper/2510.10105/) (2025)
- [RDTE-UNet: A Boundary and Detail Aware UNet for Precise Medical Image Segmentation](https://mlsanity.com/paper/2511.01328/) (2025)
- [Toward a benchmark for CTR prediction in online advertising: datasets, evaluation protocols and perspectives](https://mlsanity.com/paper/2512.01179/) (2025)
- [Deferred is Better: A Framework for Multi-Granularity Deferred Interaction of Heterogeneous Features](https://mlsanity.com/paper/2603.12586/) (2026)
- [Intelligent Approval of Access Control Flow in Office Automation Systems via Relational Modeling](https://mlsanity.com/paper/2604.11040/) (2026)
- [Revisiting Content-Based Music Recommendation: Efficient Feature Aggregation from Large-Scale Music Models](https://mlsanity.com/paper/2604.20847/) (2026)
- [Beyond Interaction Capacity: Estimator Scaling with Recursive Models for CTR Prediction](https://mlsanity.com/paper/2609.37905/) (2026)

## Abstract (arXiv)

Learning effective high-order feature interactions is very crucial in the CTR prediction task. However, it is very time-consuming to calculate high-order feature interactions with massive features in online e-commerce platforms. Most existing methods manually design a maximal order and further filter out the useless interactions from them. Although they reduce the high computational costs caused by the exponential growth of high-order feature combinations, they still suffer from the degradation of model capability due to the suboptimal learning of the restricted feature orders. The solution to maintain the model capability and meanwhile keep it efficient is a technical challenge, which has not been adequately addressed. To address this issue, we propose an adaptive feature interaction learning model, named as EulerNet, in which the feature interactions are learned in a complex vector space by conducting space mapping according to Euler's formula. EulerNet converts the exponential powers of feature interactions into simple linear combinations of the modulus and phase of the complex features, making it possible to adaptively learn the high-order feature interactions in an efficient way. Furthermore, EulerNet incorporates the implicit and explicit feature interactions into a unified architecture, which achieves the mutual enhancement and largely boosts the model capabilities. Such a network can be fully learned from data, with no need of pre-designed form or order for feature interactions. Extensive experiments conducted on three public datasets have demonstrated the effectiveness and efficiency of our approach. Our code is available at: https://github.com/RUCAIBox/EulerNet.