# Encoding word order in complex embeddings

arXiv: https://arxiv.org/abs/1912.12333 · HTML: https://mlsanity.com/paper/1912.12333/

## Key facts

- Cited by 53 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Dec 2019.

## Built on by (most cited first, 30 of 53)

- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Conditional Positional Encodings for Vision Transformers](https://mlsanity.com/paper/2102.10882/) (2021)
- [A Survey of Transformers](https://mlsanity.com/paper/2106.04554/) (2021)
- [LIFT: Language-Interfaced Fine-Tuning for Non-Language Machine Learning Tasks](https://mlsanity.com/paper/2206.06565/) (2022)
- [Investigating the Limitations of Transformers with Simple Arithmetic Tasks](https://mlsanity.com/paper/2102.13019/) (2021)
- [Self-Attention Networks Can Process Bounded Hierarchical Languages](https://mlsanity.com/paper/2105.11115/) (2021)
- [Position Information in Transformers: An Overview](https://mlsanity.com/paper/2102.11090/) (2021)
- [Learnable Fourier Features for Multi-Dimensional Spatial Positional Encoding](https://mlsanity.com/paper/2106.02795/) (2021)
- [Pre-trained Language Models in Biomedical Domain: A Systematic Survey](https://mlsanity.com/paper/2110.05006/) (2021)
- [Your Transformer May Not be as Powerful as You Expect](https://mlsanity.com/paper/2205.13401/) (2022)
- [CAPE: Encoding Relative Positions with Continuous Augmented Positional Embeddings](https://mlsanity.com/paper/2106.03143/) (2021)
- [Fourier Position Embedding: Enhancing Attention's Periodic Extension for Length Generalization](https://mlsanity.com/paper/2412.17739/) (2024)
- [Relative Positional Encoding for Transformers with Linear Complexity](https://mlsanity.com/paper/2105.08399/) (2021)
- [Rethinking Positional Encoding](https://mlsanity.com/paper/2107.02561/) (2021)
- [Length Extrapolation of Transformers: A Survey from the Perspective of Positional Encoding](https://mlsanity.com/paper/2312.17044/) (2023)
- [Dynamic Memory based Attention Network for Sequential Recommendation](https://mlsanity.com/paper/2102.09269/) (2021)
- [Trading Positional Complexity vs. Deepness in Coordinate Networks](https://mlsanity.com/paper/2205.08987/) (2022)
- [Building Blocks for a Complex-Valued Transformer Architecture](https://mlsanity.com/paper/2306.09827/) (2023)
- [3D-RPE: Enhancing Long-Context Modeling Through 3D Rotary Position Encoding](https://mlsanity.com/paper/2406.09897/) (2024)
- [Open Challenges and Opportunities in Federated Foundation Models Towards Biomedical Healthcare](https://mlsanity.com/paper/2405.06784/) (2024)
- [TE-ESN: Time Encoding Echo State Network for Prediction Based on Irregularly Sampled Time Series Data](https://mlsanity.com/paper/2105.00412/) (2021)
- [Algebraic Positional Encodings](https://mlsanity.com/paper/2312.16045/) (2023)
- [Decoupling the "What" and "Where" With Polar Coordinate Positional Embeddings](https://mlsanity.com/paper/2509.10534/) (2025)
- [Monotonic Location Attention for Length Generalization](https://mlsanity.com/paper/2305.20019/) (2023)
- [BodyGen: Advancing Towards Efficient Embodiment Co-Design](https://mlsanity.com/paper/2503.00533/) (2025)
- [Learning Positional Embeddings for Coordinate-MLPs](https://mlsanity.com/paper/2112.11577/) (2021)
- [Group Representational Position Encoding](https://mlsanity.com/paper/2512.07805/) (2025)
- [Expanding Expressivity in Transformer Models with M"obiusAttention](https://mlsanity.com/paper/2409.12175/) (2024)
- [Conformer-based End-to-end Speech Recognition With Rotary Position Embedding](https://mlsanity.com/paper/2107.05907/) (2021)
- [Ultra-high Resolution Image Segmentation via Locality-aware Context Fusion and Alternating Local Enhancement](https://mlsanity.com/paper/2109.02580/) (2021)

## Abstract (arXiv)

Sequential word order is important when processing text. Currently, neural networks (NNs) address this by modeling word position using position embeddings. The problem is that position embeddings capture the position of individual words, but not the ordered relationship (e.g., adjacency or precedence) between individual word positions. We present a novel and principled solution for modeling both the global absolute positions of words and their order relationships. Our solution generalizes word embeddings, previously defined as independent vectors, to continuous word functions over a variable (position). The benefit of continuous functions over variable positions is that word representations shift smoothly with increasing positions. Hence, word representations in different positions can correlate with each other in a continuous function. The general solution of these functions is extended to complex-valued domain due to richer representations. We extend CNN, RNN and Transformer NNs to complex-valued versions to incorporate our complex embedding (we make all code available). Experiments on text classification, machine translation and language modeling show gains over both classical word embeddings and position-enriched word embeddings. To our knowledge, this is the first work in NLP to link imaginary numbers in complex-valued representations to concrete meanings (i.e., word order).