# Expand-and-Cluster: Parameter Recovery of Neural Networks

arXiv: https://arxiv.org/abs/2304.12794 · HTML: https://mlsanity.com/paper/2304.12794/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Apr 2023.

## Built on by (most cited first, 15 of 15)

- [Symmetry in Neural Network Parameter Spaces](https://mlsanity.com/paper/2506.13018/) (2025)
- [Identifiability of Deep Polynomial Neural Networks](https://mlsanity.com/paper/2506.17093/) (2025)
- [Polynomial Time Cryptanalytic Extraction of Deep Neural Networks in the Hard-Label Setting](https://mlsanity.com/paper/2410.05750/) (2024)
- [Learning Gaussian Multi-Index Models with Gradient Flow: Time Complexity and Directional Convergence](https://mlsanity.com/paper/2411.08798/) (2024)
- [Beyond Slow Signs in High-fidelity Model Extraction](https://mlsanity.com/paper/2406.10011/) (2024)
- [When Are Bias-Free ReLU Networks Effectively Linear Networks?](https://mlsanity.com/paper/2406.12615/) (2024)
- [Flat Channels to Infinity in Neural Loss Landscapes](https://mlsanity.com/paper/2506.14951/) (2025)
- [Navigating the Deep: End-to-End Extraction on Deep Neural Networks](https://mlsanity.com/paper/2506.17047/) (2025)
- [Make Haste Slowly: A Theory of Emergent Structured Mixed Selectivity in Feature Learning ReLU Networks](https://mlsanity.com/paper/2503.06181/) (2025)
- [Is the Hard-Label Cryptanalytic Model Extraction Really Polynomial?](https://mlsanity.com/paper/2510.06692/) (2025)
- [Measuring and Controlling Solution Degeneracy across Task-Trained Recurrent Neural Networks](https://mlsanity.com/paper/2410.03972/) (2024)
- [Data Augmentation Techniques to Reverse-Engineer Neural Network Weights from Input-Output Queries](https://mlsanity.com/paper/2511.20312/) (2025)
- [The Role of Symmetry in Optimizing Overparameterized Networks](https://mlsanity.com/paper/2604.25150/) (2026)
- [Teacher Geometry Shapes Learnability in Teacher-Student Networks](https://mlsanity.com/paper/2609.09595/) (2026)
- [End-to-End Hard-Label Cryptanalytic Model Extraction Using Efficient Sign Recovery](https://mlsanity.com/paper/2609.21941/) (2026)

## Abstract (arXiv)

Can we identify the weights of a neural network by probing its input-output mapping? At first glance, this problem seems to have many solutions because of permutation, overparameterisation and activation function symmetries. Yet, we show that the incoming weight vector of each neuron is identifiable up to sign or scaling, depending on the activation function. Our novel method 'Expand-and-Cluster' can identify layer sizes and weights of a target network for all commonly used activation functions. Expand-and-Cluster consists of two phases: (i) to relax the non-convex optimisation problem, we train multiple overparameterised student networks to best imitate the target function; (ii) to reverse engineer the target network's weights, we employ an ad-hoc clustering procedure that reveals the learnt weight vectors shared between students -- these correspond to the target weight vectors. We demonstrate successful weights and size recovery of trained shallow and deep networks with less than 10% overhead in the layer size and describe an `ease-of-identifiability' axis by analysing 150 synthetic problems of variable difficulty.