# A General Framework For Proving The Equivariant Strong Lottery Ticket Hypothesis

arXiv: https://arxiv.org/abs/2206.04270 · HTML: https://mlsanity.com/paper/2206.04270/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 28 references that are in the map.
- First posted on arXiv 9 Jun 2022.

## Builds on (most cited first, 14 of 14)

- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [What is the State of Neural Network Pruning?](https://mlsanity.com/paper/2003.03033/) (2020)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [What's Hidden in a Randomly Weighted Neural Network?](https://mlsanity.com/paper/1911.13299/) (2019)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Proving the Lottery Ticket Hypothesis: Pruning is All You Need](https://mlsanity.com/paper/2002.00585/) (2020)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Optimal Lottery Tickets via SubsetSum: Logarithmic Over-Parameterization is Sufficient](https://mlsanity.com/paper/2006.07990/) (2020)
- [Logarithmic Pruning is All You Need](https://mlsanity.com/paper/2006.12156/) (2020)
- [On the Existence of Universal Lottery Tickets](https://mlsanity.com/paper/2111.11146/) (2021)
- [Most Activation Functions Can Win the Lottery Without Excessive Depth](https://mlsanity.com/paper/2205.02321/) (2022)
- [Convolutional and Residual Networks Provably Contain Lottery Tickets](https://mlsanity.com/paper/2205.02343/) (2022)

## Built on by (most cited first, 19 of 19)

- [Why Random Pruning Is All We Need to Start Sparse](https://mlsanity.com/paper/2210.02412/) (2022)
- [Proving Linear Mode Connectivity of Neural Networks via Optimal Transport](https://mlsanity.com/paper/2310.19103/) (2023)
- [A Survey of Lottery Ticket Hypothesis](https://mlsanity.com/paper/2403.04861/) (2024)
- [Spectral Graph Pruning Against Over-Squashing and Over-Smoothing](https://mlsanity.com/paper/2404.04612/) (2024)
- [Masks, Signs, And Learning Rate Rewinding](https://mlsanity.com/paper/2402.19262/) (2024)
- [Mask in the Mirror: Implicit Sparsification](https://mlsanity.com/paper/2408.09966/) (2024)
- [Successfully Applying Lottery Ticket Hypothesis to Diffusion Model](https://mlsanity.com/paper/2310.18823/) (2023)
- [Strong Lottery Ticket Hypothesis with varepsilon--perturbation](https://mlsanity.com/paper/2210.16589/) (2022)
- [Polynomially Over-Parameterized Convolutional Neural Networks Contain Structured Strong Winning Lottery Tickets](https://mlsanity.com/paper/2311.09858/) (2023)
- [On the Sparsity of the Strong Lottery Ticket Hypothesis](https://mlsanity.com/paper/2410.14754/) (2024)
- [The Strong Lottery Ticket Hypothesis for Multi-Head Attention Mechanisms](https://mlsanity.com/paper/2511.04217/) (2025)
- [Considering Layerwise Importance in the Lottery Ticket Hypothesis](https://mlsanity.com/paper/2302.11244/) (2023)
- [Pruning neural network models for gene regulatory dynamics using data and domain knowledge](https://mlsanity.com/paper/2403.04805/) (2024)
- [Cyclic Sparse Training: Is it Enough?](https://mlsanity.com/paper/2406.02773/) (2024)
- [Quantization vs Pruning: Insights from the Strong Lottery Ticket Hypothesis](https://mlsanity.com/paper/2508.11020/) (2025)
- [Online Algorithms with Limited Data Retention](https://mlsanity.com/paper/2404.10997/) (2024)
- [Investigating the Lottery Ticket Hypothesis for Variational Quantum Circuits](https://mlsanity.com/paper/2509.11190/) (2025)
- [Structured vs. Unstructured Pruning: An Exponential Gap](https://mlsanity.com/paper/2603.02234/) (2026)
- [A Unified Framework for Quantized and Continuous Strong Lottery Tickets](https://mlsanity.com/paper/2607.03860/) (2026)

## Abstract (arXiv)

The Strong Lottery Ticket Hypothesis (SLTH) stipulates the existence of a subnetwork within a sufficiently overparameterized (dense) neural network that -- when initialized randomly and without any training -- achieves the accuracy of a fully trained target network. Recent works by Da Cunha et. al 2022; Burkholz 2022 demonstrate that the SLTH can be extended to translation equivariant networks -- i.e. CNNs -- with the same level of overparametrization as needed for the SLTs in dense networks. However, modern neural networks are capable of incorporating more than just translation symmetry, and developing general equivariant architectures such as rotation and permutation has been a powerful design principle. In this paper, we generalize the SLTH to functions that preserve the action of the group G -- i.e. G-equivariant network -- and prove, with high probability, that one can approximate any G-equivariant network of fixed width and depth by pruning a randomly initialized overparametrized G-equivariant network to a G-equivariant subnetwork. We further prove that our prescribed overparametrization scheme is optimal and provides a lower bound on the number of effective parameters as a function of the error tolerance. We develop our theory for a large range of groups, including subgroups of the Euclidean E(2) and Symmetric group G ≤ Sn -- allowing us to find SLTs for MLPs, CNNs, E(2)-steerable CNNs, and permutation equivariant networks as specific instantiations of our unified framework. Empirically, we verify our theory by pruning overparametrized E(2)-steerable CNNs, k-order GNNs, and message passing GNNs to match the performance of trained target networks.