# Mitigating Over-smoothing in Transformers via Regularized Nonlocal Functionals

arXiv: https://arxiv.org/abs/2312.00751 · HTML: https://mlsanity.com/paper/2312.00751/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 71 references that are in the map.
- First posted on arXiv 1 Dec 2023.

## Builds on (most cited first, 30 of 50)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Training data-efficient image transformers & distillation through attention](https://mlsanity.com/paper/2012.12877/) (2020)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [SQuAD: 100,000+ Questions for Machine Comprehension of Text](https://mlsanity.com/paper/1606.05250/) (2016)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [Benchmarking Neural Network Robustness to Common Corruptions and Perturbations](https://mlsanity.com/paper/1903.12261/) (2019)
- [XLNet: Generalized Autoregressive Pretraining for Language Understanding](https://mlsanity.com/paper/1906.08237/) (2019)
- [SimCSE: Simple Contrastive Learning of Sentence Embeddings](https://mlsanity.com/paper/2104.08821/) (2021)
- [Conformer: Convolution-augmented Transformer for Speech Recognition](https://mlsanity.com/paper/2005.08100/) (2020)
- [Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context](https://mlsanity.com/paper/1901.02860/) (2019)
- [The Many Faces of Robustness: A Critical Analysis of Out-of-Distribution Generalization](https://mlsanity.com/paper/2006.16241/) (2020)
- [ViViT: A Video Vision Transformer](https://mlsanity.com/paper/2103.15691/) (2021)
- [Generating Long Sequences with Sparse Transformers](https://mlsanity.com/paper/1904.10509/) (2019)
- [Decision Transformer: Reinforcement Learning via Sequence Modeling](https://mlsanity.com/paper/2106.01345/) (2021)
- [Point Transformer](https://mlsanity.com/paper/2011.00931/) (2020)
- [Natural Adversarial Examples](https://mlsanity.com/paper/1907.07174/) (2019)
- [Video Swin Transformer](https://mlsanity.com/paper/2106.13230/) (2021)
- [Semantic Understanding of Scenes through the ADE20K Dataset](https://mlsanity.com/paper/1608.05442/) (2016)
- [BERT Rediscovers the Classical NLP Pipeline](https://mlsanity.com/paper/1905.05950/) (2019)
- [What Does BERT Look At? An Analysis of BERT's Attention](https://mlsanity.com/paper/1906.04341/) (2019)
- [Analyzing Multi-Head Self-Attention: Specialized Heads Do the Heavy Lifting, the Rest Can Be Pruned](https://mlsanity.com/paper/1905.09418/) (2019)

## Built on by (most cited first, 30 of 32)

- [Stop Looking for Important Tokens in Multimodal Language Models: Duplication Matters More](https://mlsanity.com/paper/2502.11494/) (2025)
- [The Hidden Attention of Mamba Models](https://mlsanity.com/paper/2403.01590/) (2024)
- [Value Residual Learning](https://mlsanity.com/paper/2410.17897/) (2024)
- [Table Foundation Models: on knowledge pre-training for tabular learning](https://mlsanity.com/paper/2505.14415/) (2025)
- [UniGuardian: A Unified Defense for Detecting Prompt Injection, Backdoor Attacks and Adversarial Attacks in Large Language Models](https://mlsanity.com/paper/2502.13141/) (2025)
- [Graph Convolutions Enrich the Self-Attention in Transformers!](https://mlsanity.com/paper/2312.04234/) (2023)
- [Frequency-Dynamic Attention Modulation for Dense Prediction](https://mlsanity.com/paper/2507.12006/) (2025)
- [Multistability of Self-Attention Dynamics in Transformers](https://mlsanity.com/paper/2511.11553/) (2025)
- [Unveiling the Hidden Structure of Self-Attention via Kernel Principal Component Analysis](https://mlsanity.com/paper/2406.13762/) (2024)
- [Elliptical Attention](https://mlsanity.com/paper/2406.13770/) (2024)
- [CR-Net: Scaling Parameter-Efficient Training with Cross-Layer Low-Rank Structure](https://mlsanity.com/paper/2509.18993/) (2025)
- [Reconstructing KV Caches with Cross-layer Fusion For Enhanced Transformers](https://mlsanity.com/paper/2512.03870/) (2025)
- [ANCRe: Adaptive Neural Connection Reassignment for Efficient Depth Scaling](https://mlsanity.com/paper/2602.09009/) (2026)
- [MOOSComp: Improving Lightweight Long-Context Compressor via Mitigating Over-Smoothing and Incorporating Outlier Scores](https://mlsanity.com/paper/2504.16786/) (2025)
- [SoftHGNN: Soft Hypergraph Neural Networks for General Visual Recognition](https://mlsanity.com/paper/2505.15325/) (2025)
- [Improving Model Representation and Reducing KV Cache via Skip Connections with First Value Heads](https://mlsanity.com/paper/2510.16807/) (2025)
- [Transformer Meets Twicing: Harnessing Unattended Residual Information](https://mlsanity.com/paper/2503.00687/) (2025)
- [A Gravity-informed Spatiotemporal Transformer for Human Activity Intensity Prediction](https://mlsanity.com/paper/2506.13678/) (2025)
- [Wavy Transformer](https://mlsanity.com/paper/2508.12787/) (2025)
- [ECG-Soup: Harnessing Multi-Layer Synergy for ECG Foundation Models](https://mlsanity.com/paper/2509.00102/) (2025)
- [Hierarchical Self-Attention: Generalizing Neural Attention Mechanics to Multi-Scale Problems](https://mlsanity.com/paper/2509.15448/) (2025)
- [Do Value Vectors in Deep Layers Need Context from the Residual Stream?](https://mlsanity.com/paper/2606.02780/) (2026)
- [Lightweight Adaptive Feature Composition for Heterogeneous Downstream Adaptation of Wireless Foundation Models](https://mlsanity.com/paper/2606.10277/) (2026)
- [Not All Attention Heads Contribute to Critical Visual Token Selection: Head-Aware Pruning Matters More](https://mlsanity.com/paper/2608.25332/) (2026)
- [Transformer-based Graph Neural Networks for Battery Range Prediction in AIoT Battery-Swap Services](https://mlsanity.com/paper/2407.16115/) (2024)
- [Revisiting Transformers with Insights from Image Filtering and Boosting](https://mlsanity.com/paper/2506.10371/) (2025)
- [MSTN: A Lightweight and Fast Model for General TimeSeries Analysis](https://mlsanity.com/paper/2511.20577/) (2025)
- [Analogies between Transformer Layers and Power Method](https://mlsanity.com/paper/2605.25619/) (2026)
- [FAiT: Frequency-Aware Inverted Transformer for Multivariate Time Series Forecasting](https://mlsanity.com/paper/2606.01306/) (2026)
- [The Devil is in the Spectrum: Mitigating Representation Collapse in LLMs via Topologically Regularized Side-Path](https://mlsanity.com/paper/2607.20484/) (2026)

## Abstract (arXiv)

Transformers have achieved remarkable success in a wide range of natural language processing and computer vision applications. However, the representation capacity of a deep transformer model is degraded due to the over-smoothing issue in which the token representations become identical when the model's depth grows. In this work, we show that self-attention layers in transformers minimize a functional which promotes smoothness, thereby causing token uniformity. We then propose a novel regularizer that penalizes the norm of the difference between the smooth output tokens from self-attention and the input tokens to preserve the fidelity of the tokens. Minimizing the resulting regularized energy functional, we derive the Neural Transformer with a Regularized Nonlocal Functional (NeuTRENO), a novel class of transformer models that can mitigate the over-smoothing issue. We empirically demonstrate the advantages of NeuTRENO over the baseline transformers and state-of-the-art methods in reducing the over-smoothing of token representations on various practical tasks, including object classification, image segmentation, and language modeling.