# Canonical locks that encode part-whole hierarchies

arXiv: https://arxiv.org/abs/2609.26046 · HTML: https://mlsanity.com/paper/2609.26046/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 39 references that are in the map.
- First posted on arXiv 22 Sep 2026.

## Builds on (most cited first, 20 of 20)

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- [Dynamic Routing Between Capsules](https://mlsanity.com/paper/1710.09829/) (2017)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [The Forward-Forward Algorithm: Some Preliminary Investigations](https://mlsanity.com/paper/2212.13345/) (2022)
- [How to represent part-whole hierarchies in a neural network](https://mlsanity.com/paper/2102.12627/) (2021)
- [LieTransformer: Equivariant self-attention for Lie Groups](https://mlsanity.com/paper/2012.10885/) (2020)
- [Grammar as a Foreign Language](https://mlsanity.com/paper/1412.7449/) (2014)
- [A Modern Self-Referential Weight Matrix That Learns to Modify Itself](https://mlsanity.com/paper/2202.05780/) (2022)
- [Complex-Valued Autoencoders for Object Discovery](https://mlsanity.com/paper/2204.02075/) (2022)
- [Artificial Kuramoto Oscillatory Neurons](https://mlsanity.com/paper/2410.13821/) (2024)
- [Rotating Features for Object Discovery](https://mlsanity.com/paper/2306.00600/) (2023)
- [Interpretable part-whole hierarchies and conceptual-semantic relationships in neural networks](https://mlsanity.com/paper/2203.03282/) (2022)
- [On Occlusions in Video Action Detection: Benchmark Datasets And Training Recipes](https://mlsanity.com/paper/2410.19553/) (2024)
- [Krause Synchronization Transformers](https://mlsanity.com/paper/2602.11534/) (2026)
- [Asynchronous Perception Machine For Efficient Test-Time-Training](https://mlsanity.com/paper/2410.20535/) (2024)

## Abstract (arXiv)

One of the challenges in representational learning is how to encode part-whole hierarchies in a neural net. Prior works rely on flattening tree-like structures into string-like sequences and training a sequence-to-sequence model via autoregression. While such a representation works for parse-trees in NLP, it is not entirely clear how to make it work for images. Thus, we propose a geometric primitive called canonical locks. The key idea is that parts/wholes can be modelled as higher-dimensional vectors (d ≥ 4), and information can be encoded in their relative phase differences. Inductively, the net consists of positionally-bound bottom-up and top-down neural fields, which drive each other to achieve a state of thermal equilibrium. Additionally, we show the existence of a few symmetrical configurations in the net. The computational iterations taken to break these symmetries depend on the angle between parts/wholes arranged on a disk (or more precisely a ring) in higher dimensions. It also appears to have connections to the psychological phenomenon of mental rotation.