# Multi-manifold Attention for Vision Transformers

arXiv: https://arxiv.org/abs/2207.08569 · HTML: https://mlsanity.com/paper/2207.08569/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 49 of its 72 references that are in the map.
- First posted on arXiv 18 Jul 2022.

## Builds on (most cited first, 30 of 49)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks](https://mlsanity.com/paper/1905.11946/) (2019)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [mixup: Beyond Empirical Risk Minimization](https://mlsanity.com/paper/1710.09412/) (2017)
- [Training data-efficient image transformers & distillation through attention](https://mlsanity.com/paper/2012.12877/) (2020)
- [A ConvNet for the 2020s](https://mlsanity.com/paper/2201.03545/) (2022)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Pyramid Vision Transformer: A Versatile Backbone for Dense Prediction without Convolutions](https://mlsanity.com/paper/2102.12122/) (2021)
- [Reformer: The Efficient Transformer](https://mlsanity.com/paper/2001.04451/) (2020)
- [Rethinking Attention with Performers](https://mlsanity.com/paper/2009.14794/) (2020)
- [Accurate, Large Minibatch SGD: Training ImageNet in 1 Hour](https://mlsanity.com/paper/1706.02677/) (2017)
- [Linformer: Self-Attention with Linear Complexity](https://mlsanity.com/paper/2006.04768/) (2020)
- [Swin Transformer V2: Scaling Up Capacity and Resolution](https://mlsanity.com/paper/2111.09883/) (2021)
- [Tokens-to-Token ViT: Training Vision Transformers from Scratch on ImageNet](https://mlsanity.com/paper/2101.11986/) (2021)
- [Semantic Understanding of Scenes through the ADE20K Dataset](https://mlsanity.com/paper/1608.05442/) (2016)
- [Transformers in Vision: A Survey](https://mlsanity.com/paper/2101.01169/) (2021)
- [Transformer in Transformer](https://mlsanity.com/paper/2103.00112/) (2021)
- [Twins: Revisiting the Design of Spatial Attention in Vision Transformers](https://mlsanity.com/paper/2104.13840/) (2021)
- [CSWin Transformer: A General Vision Transformer Backbone with Cross-Shaped Windows](https://mlsanity.com/paper/2107.00652/) (2021)
- [MetaFormer Is Actually What You Need for Vision](https://mlsanity.com/paper/2111.11418/) (2021)
- [ConViT: Improving Vision Transformers with Soft Convolutional Inductive Biases](https://mlsanity.com/paper/2103.10697/) (2021)
- [Bag of Tricks for Image Classification with Convolutional Neural Networks](https://mlsanity.com/paper/1812.01187/) (2018)
- [Conditional Positional Encodings for Vision Transformers](https://mlsanity.com/paper/2102.10882/) (2021)
- [AdaViT: Adaptive Tokens for Efficient Vision Transformer](https://mlsanity.com/paper/2112.07658/) (2021)
- [Rethinking Spatial Dimensions of Vision Transformers](https://mlsanity.com/paper/2103.16302/) (2021)
- [Vision Transformer with Deformable Attention](https://mlsanity.com/paper/2201.00520/) (2022)
- [DeepViT: Towards Deeper Vision Transformer](https://mlsanity.com/paper/2103.11886/) (2021)

## Built on by (most cited first, 12 of 12)

- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [Position: Beyond Euclidean -- Foundation Models Should Embrace Non-Euclidean Geometries](https://mlsanity.com/paper/2504.08896/) (2025)
- [KANICE: Kolmogorov-Arnold Networks with Interactive Convolutional Elements](https://mlsanity.com/paper/2410.17172/) (2024)
- [Structure-Preserving Transformers for Sequences of SPD Matrices](https://mlsanity.com/paper/2309.07579/) (2023)
- [A Random Ensemble of Encrypted Vision Transformers for Adversarially Robust Defense](https://mlsanity.com/paper/2402.07183/) (2024)
- [A Manifold Representation of the Key in Vision Transformers](https://mlsanity.com/paper/2402.00534/) (2024)
- [Leader-Follower Neural Networks with Local Error Signals Inspired by Complex Collectives](https://mlsanity.com/paper/2310.07885/) (2023)
- [Geometric sparsification in recurrent neural networks](https://mlsanity.com/paper/2406.06290/) (2024)
- [Context-Aware Token Selection and Packing for Enhanced Vision Transformer](https://mlsanity.com/paper/2410.23608/) (2024)
- [GeoNorm: Unify Pre-Norm and Post-Norm with Geodesic Optimization](https://mlsanity.com/paper/2601.22095/) (2026)
- [Automatic Classification of Sleep Stages from EEG Signals Using Riemannian Metrics and Transformer Networks](https://mlsanity.com/paper/2410.19819/) (2024)
- [Image Classification via Random Dilated Convolution with Multi-Branch Feature Extraction and Context Excitation](https://mlsanity.com/paper/2604.25188/) (2026)

## Abstract (arXiv)

Vision Transformers are very popular nowadays due to their state-of-the-art performance in several computer vision tasks, such as image classification and action recognition. Although their performance has been greatly enhanced through highly descriptive patch embeddings and hierarchical structures, there is still limited research on utilizing additional data representations so as to refine the selfattention map of a Transformer. To address this problem, a novel attention mechanism, called multi-manifold multihead attention, is proposed in this work to substitute the vanilla self-attention of a Transformer. The proposed mechanism models the input space in three distinct manifolds, namely Euclidean, Symmetric Positive Definite and Grassmann, thus leveraging different statistical and geometrical properties of the input for the computation of a highly descriptive attention map. In this way, the proposed attention mechanism can guide a Vision Transformer to become more attentive towards important appearance, color and texture features of an image, leading to improved classification and segmentation results, as shown by the experimental results on well-known datasets.