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

TL;DR

The paper addresses learning from complex-valued signals where both amplitude $A$ and phase $\Phi$ matter, noting that standard attention often relies on magnitude and ignores interference. It proposes the Holographic Transformer, whose interference-aware self-attention rotates values by phase differences $\Delta\phi_{ij}$ and aggregates via $H_i=\sum_j \alpha_{ij} V_j \exp(j\Delta\phi_{ij})$, together with a dual-headed decoder. Theoretical contributions establish reduction to standard attention when $\Delta\phi_{ij}=0$, global phase equivariance, energy bounds, a weighted Fréchet mean / MLE interpretation, Lipschitz stability to phase perturbations, and a mechanism to prevent phase collapse. Empirically, the method achieves state-of-the-art accuracy and robustness on PolSAR classification and wireless channel prediction, highlighting the practical value of enforcing physical coherence in complex-valued learning.

Abstract

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.

Figures (1)

• Figure 1: Holographic Transformer Architecture