# Hierarchical Self-Attention: Generalizing Neural Attention Mechanics to Multi-Scale Problems

arXiv: https://arxiv.org/abs/2509.15448 · HTML: https://mlsanity.com/paper/2509.15448/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 78 of its 100 references that are in the map.
- First posted on arXiv 18 Sep 2025.

## Builds on (most cited first, 30 of 78)

- [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)
- [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)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [Training data-efficient image transformers & distillation through attention](https://mlsanity.com/paper/2012.12877/) (2020)
- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [Longformer: The Long-Document Transformer](https://mlsanity.com/paper/2004.05150/) (2020)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Character-level Convolutional Networks for Text Classification](https://mlsanity.com/paper/1509.01626/) (2015)
- [ViLBERT: Pretraining Task-Agnostic Visiolinguistic Representations for Vision-and-Language Tasks](https://mlsanity.com/paper/1908.02265/) (2019)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [Is Space-Time Attention All You Need for Video Understanding?](https://mlsanity.com/paper/2102.05095/) (2021)
- [ViViT: A Video Vision Transformer](https://mlsanity.com/paper/2103.15691/) (2021)
- [Rethinking Attention with Performers](https://mlsanity.com/paper/2009.14794/) (2020)
- [Generating Long Sequences with Sparse Transformers](https://mlsanity.com/paper/1904.10509/) (2019)
- [Point Transformer](https://mlsanity.com/paper/2011.00931/) (2020)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [VisualBERT: A Simple and Performant Baseline for Vision and Language](https://mlsanity.com/paper/1908.03557/) (2019)
- [ViLT: Vision-and-Language Transformer Without Convolution or Region Supervision](https://mlsanity.com/paper/2102.03334/) (2021)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Multimodal Machine Learning: A Survey and Taxonomy](https://mlsanity.com/paper/1705.09406/) (2017)
- [PCT: Point cloud transformer](https://mlsanity.com/paper/2012.09688/) (2020)
- [Recipe for a General, Powerful, Scalable Graph Transformer](https://mlsanity.com/paper/2205.12454/) (2022)
- [Vector-quantized Image Modeling with Improved VQGAN](https://mlsanity.com/paper/2110.04627/) (2021)

## Built on by (most cited first, 1 of 1)

- [Overlapping Schwarz Attention: Hierarchical Attention via Domain Decomposition](https://mlsanity.com/paper/2606.18525/) (2026)

## Abstract (arXiv)

Transformers and their attention mechanism have been revolutionary in the field of Machine Learning. While originally proposed for the language data, they quickly found their way to the image, video, graph, etc. data modalities with various signal geometries. Despite this versatility, generalizing the attention mechanism to scenarios where data is presented at different scales from potentially different modalities is not straightforward. The attempts to incorporate hierarchy and multi-modality within transformers are largely based on ad hoc heuristics, which are not seamlessly generalizable to similar problems with potentially different structures. To address this problem, in this paper, we take a fundamentally different approach: we first propose a mathematical construct to represent multi-modal, multi-scale data. We then mathematically derive the neural attention mechanics for the proposed construct from the first principle of entropy minimization. We show that the derived formulation is optimal in the sense of being the closest to the standard Softmax attention while incorporating the inductive biases originating from the hierarchical/geometric information of the problem. We further propose an efficient algorithm based on dynamic programming to compute our derived attention mechanism. By incorporating it within transformers, we show that the proposed hierarchical attention mechanism not only can be employed to train transformer models in hierarchical/multi-modal settings from scratch, but it can also be used to inject hierarchical information into classical, pre-trained transformer models post training, resulting in more efficient models in zero-shot manner.