# Unlocking Symbol-Level Precoding Efficiency Through Tensor Equivariant Neural Network

arXiv: https://arxiv.org/abs/2510.02108 · HTML: https://mlsanity.com/paper/2510.02108/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 54 references that are in the map.
- First posted on arXiv 2 Oct 2025.

## Builds on (most cited first, 28 of 28)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [CBAM: Convolutional Block Attention Module](https://mlsanity.com/paper/1807.06521/) (2018)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Are Transformers universal approximators of sequence-to-sequence functions?](https://mlsanity.com/paper/1912.10077/) (2019)
- [Pure Transformers are Powerful Graph Learners](https://mlsanity.com/paper/2207.02505/) (2022)
- [Universal Invariant and Equivariant Graph Neural Networks](https://mlsanity.com/paper/1905.04943/) (2019)
- [Deep Models of Interactions Across Sets](https://mlsanity.com/paper/1803.02879/) (2018)
- [RE-MIMO: Recurrent and Permutation Equivariant Neural MIMO Detection](https://mlsanity.com/paper/2007.00140/) (2020)
- [Interference Exploitation-based Hybrid Precoding with Robustness Against Phase Errors](https://mlsanity.com/paper/1810.04490/) (2018)
- [Interference Exploitation Precoding for Multi-Level Modulations: Closed-Form Solutions](https://mlsanity.com/paper/1811.03289/) (2018)
- [An Unsupervised Learning-Based Approach for Symbol-Level-Precoding](https://mlsanity.com/paper/2104.09214/) (2021)
- [An Unsupervised Deep Unfolding Framework for robust Symbol Level Precoding](https://mlsanity.com/paper/2111.08129/) (2021)
- [Towards Unified AI Models for MU-MIMO Communications: A Tensor Equivariance Framework](https://mlsanity.com/paper/2406.09022/) (2024)
- [Statistical CSI-Based Distributed Precoding Design for OFDM-Cooperative Multi-Satellite Systems](https://mlsanity.com/paper/2505.08038/) (2025)
- [Extract the Best, Discard the Rest: CSI Feedback with Offline Large AI Models](https://mlsanity.com/paper/2505.08566/) (2025)
- [Robust Design of Power Minimizing Symbol-Level Precoder under Channel Uncertainty](https://mlsanity.com/paper/1805.02395/) (2018)
- [Power Minimizer Symbol-Level Precoding: A Closed-Form Sub-Optimal Solution](https://mlsanity.com/paper/1807.10619/) (2018)
- [Robust SINR-Constrained Symbol-Level Multiuser Precoding with Imperfect Channel Knowledge](https://mlsanity.com/paper/1903.03371/) (2019)
- [An Approximate Solution for Symbol-Level Multiuser Precoding Using Support Recovery](https://mlsanity.com/paper/1903.03377/) (2019)
- [Minimum Symbol-Error Probability Symbol-Level Precoding with Intelligent Reflecting Surface](https://mlsanity.com/paper/2001.06840/) (2020)
- [Symbol-Level Precoding Made Practical for Multi-Level Modulations via Block-Level Rescaling](https://mlsanity.com/paper/2006.15245/) (2020)
- [Weighted MMSE Precoding for Constructive Interference Region](https://mlsanity.com/paper/2210.00167/) (2022)
- [Symbol-Level Precoding for Average SER Minimization in Multiuser MISO Systems](https://mlsanity.com/paper/2310.07436/) (2023)
- [Soft Demodulator for Symbol-Level Precoding in Coded Multiuser MISO Systems](https://mlsanity.com/paper/2310.10296/) (2023)
- [Robust Symbol-Level Precoding for Massive MIMO Communication Under Channel Aging](https://mlsanity.com/paper/2402.04685/) (2024)
- [A Tensor-Structured Approach to Dynamic Channel Prediction for Massive MIMO Systems with Temporal Non-Stationarity](https://mlsanity.com/paper/2412.06713/) (2024)
- [CovNet: Covariance Information-Assisted CSI Feedback for FDD Massive MIMO Systems](https://mlsanity.com/paper/2412.12875/) (2024)
- [Tensor-Structured Bayesian Channel Prediction for Upper Mid-Band XL-MIMO Systems](https://mlsanity.com/paper/2508.08491/) (2025)

## Abstract (arXiv)

Although symbol-level precoding (SLP) based on constructive interference (CI) exploitation offers performance gains, its high complexity remains a bottleneck. This paper addresses this challenge with an end-to-end deep learning (DL) framework with low inference complexity that leverages the structure of the optimal SLP solution in the closed-form and its inherent tensor equivariance (TE), where TE denotes that a permutation of the input induces the corresponding permutation of the output. Building upon the computationally efficient model-based formulations, as well as their known closed-form solutions, we analyze their relationship with linear precoding (LP) and investigate the corresponding optimality condition. We then construct a mapping from the problem formulation to the solution and prove its TE, based on which the designed networks reveal a specific parameter-sharing pattern that delivers low computational complexity and strong generalization. Leveraging these, we propose the backbone of the framework with an attention-based TE module, achieving linear computational complexity. Furthermore, we demonstrate that such a framework is also applicable to imperfect CSI scenarios, where we design a TE-based network to map the CSI, statistics, and symbols to auxiliary variables. Simulation results show that the proposed framework captures substantial performance gains of optimal SLP, while achieving an approximately 80-times speedup over conventional methods and maintaining strong generalization across user numbers and symbol block lengths.