# Holographic Transformers for Complex-Valued Signal Processing: Integrating Phase Interference into Self-Attention

arXiv: https://arxiv.org/abs/2509.19331 · HTML: https://mlsanity.com/paper/2509.19331/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 28 references that are in the map.
- First posted on arXiv 14 Sep 2025.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Long Range Arena: A Benchmark for Efficient Transformers](https://mlsanity.com/paper/2011.04006/) (2020)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [DCCRN: Deep Complex Convolution Recurrent Network for Phase-Aware Speech Enhancement](https://mlsanity.com/paper/2008.00264/) (2020)
- [Phase-aware Speech Enhancement with Deep Complex U-Net](https://mlsanity.com/paper/1903.03107/) (2019)
- [A Survey of Complex-Valued Neural Networks](https://mlsanity.com/paper/2101.12249/) (2021)
- [PHASEN: A Phase-and-Harmonics-Aware Speech Enhancement Network](https://mlsanity.com/paper/1911.04697/) (2019)
- [A Phase Shift Deep Neural Network for High Frequency Approximation and Wave Problems](https://mlsanity.com/paper/1909.11759/) (2019)
- [Complex Transformer: A Framework for Modeling Complex-Valued Sequence](https://mlsanity.com/paper/1910.10202/) (2019)
- [Building Blocks for a Complex-Valued Transformer Architecture](https://mlsanity.com/paper/2306.09827/) (2023)
- [Complex-valued neural networks for machine learning on non-stationary physical data](https://mlsanity.com/paper/1905.12321/) (2019)
- [Complex-valued Convolutional Neural Networks for Enhanced Radar Signal Denoising and Interference Mitigation](https://mlsanity.com/paper/2105.00929/) (2021)
- [A Spatially Separable Attention Mechanism for massive MIMO CSI Feedback](https://mlsanity.com/paper/2208.03369/) (2022)
- [Next-slot OFDM-CSI Prediction: Multi-head Self-attention or State Space Model?](https://mlsanity.com/paper/2405.11072/) (2024)
- [Steinmetz Neural Networks for Complex-Valued Data](https://mlsanity.com/paper/2409.10075/) (2024)

## Built on by (most cited first, 2 of 2)

- [Language as a Wave Phenomenon: Semantic Phase Locking and Interference in Neural Networks](https://mlsanity.com/paper/2512.01208/) (2025)
- [Phasor Agents: Oscillatory Graphs with Three-Factor Plasticity and Sleep-Staged Learning](https://mlsanity.com/paper/2601.04362/) (2026)

## Abstract (arXiv)

Complex-valued signals encode both amplitude and phase, yet most deep models treat attention as real-valued correlation, overlooking interference effects. We introduce the Holographic Transformer, a physics-inspired architecture that incorporates wave interference principles into self-attention. Holographic attention modulates interactions by relative phase and coherently superimposes values, ensuring consistency between amplitude and phase. A dual-headed decoder simultaneously reconstructs the input and predicts task outputs, preventing phase collapse when losses prioritize magnitude over phase. We demonstrate that holographic attention implements a discrete interference operator and maintains phase consistency under linear mixing. Experiments on PolSAR image classification and wireless channel prediction show strong performance, achieving high classification accuracy and F1 scores, low regression error, and increased robustness to phase perturbations. These results highlight that enforcing physical consistency in attention leads to generalizable improvements in complex-valued learning and provides a unified, physics-based framework for coherent signal modeling. The code is available at https://github.com/EonHao/Holographic-Transformers.