# Going Beyond Neural Network Feature Similarity: The Network Feature Complexity and Its Interpretation Using Category Theory

arXiv: https://arxiv.org/abs/2310.06756 · HTML: https://mlsanity.com/paper/2310.06756/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 44 of its 62 references that are in the map.
- First posted on arXiv 10 Oct 2023.

## Builds on (most cited first, 30 of 44)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [How transferable are features in deep neural networks?](https://mlsanity.com/paper/1411.1792/) (2014)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Striving for Simplicity: The All Convolutional Net](https://mlsanity.com/paper/1412.6806/) (2014)
- [SNIP: Single-shot Network Pruning based on Connection Sensitivity](https://mlsanity.com/paper/1810.02340/) (2018)
- [Linear Mode Connectivity and the Lottery Ticket Hypothesis](https://mlsanity.com/paper/1912.05671/) (2019)
- [Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs](https://mlsanity.com/paper/1802.10026/) (2018)
- [Learning Structured Sparsity in Deep Neural Networks](https://mlsanity.com/paper/1608.03665/) (2016)
- [Pruning neural networks without any data by iteratively conserving synaptic flow](https://mlsanity.com/paper/2006.05467/) (2020)
- [Git Re-Basin: Merging Models modulo Permutation Symmetries](https://mlsanity.com/paper/2209.04836/) (2022)
- [Essentially No Barriers in Neural Network Energy Landscape](https://mlsanity.com/paper/1803.00885/) (2018)
- [Insights on representational similarity in neural networks with canonical correlation](https://mlsanity.com/paper/1806.05759/) (2018)
- [The Role of Permutation Invariance in Linear Mode Connectivity of Neural Networks](https://mlsanity.com/paper/2110.06296/) (2021)
- [Identifying and attacking the saddle point problem in high-dimensional non-convex optimization](https://mlsanity.com/paper/1406.2572/) (2014)
- [Do Wide and Deep Networks Learn the Same Things? Uncovering How Neural Network Representations Vary with Width and Depth](https://mlsanity.com/paper/2010.15327/) (2020)
- [Convergent Learning: Do different neural networks learn the same representations?](https://mlsanity.com/paper/1511.07543/) (2015)
- [Quantized Convolutional Neural Networks for Mobile Devices](https://mlsanity.com/paper/1512.06473/) (2015)
- [Ternary Weight Networks](https://mlsanity.com/paper/1605.04711/) (2016)
- [Compressing Deep Convolutional Networks using Vector Quantization](https://mlsanity.com/paper/1412.6115/) (2014)
- [Data-free parameter pruning for Deep Neural Networks](https://mlsanity.com/paper/1507.06149/) (2015)
- [Rethinking the Smaller-Norm-Less-Informative Assumption in Channel Pruning of Convolution Layers](https://mlsanity.com/paper/1802.00124/) (2018)
- [Generalized Shape Metrics on Neural Representations](https://mlsanity.com/paper/2110.14739/) (2021)
- [Convolutional neural networks with low-rank regularization](https://mlsanity.com/paper/1511.06067/) (2015)
- [Topology and Geometry of Half-Rectified Network Optimization](https://mlsanity.com/paper/1611.01540/) (2016)
- [DIFFormer: Scalable (Graph) Transformers Induced by Energy Constrained Diffusion](https://mlsanity.com/paper/2301.09474/) (2023)
- [AdderNet: Do We Really Need Multiplications in Deep Learning?](https://mlsanity.com/paper/1912.13200/) (2019)

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

- [Mechanistic Interpretability for AI Safety -- A Review](https://mlsanity.com/paper/2404.14082/) (2024)
- [Interpreting the linear structure of vision-language model embedding spaces](https://mlsanity.com/paper/2504.11695/) (2025)
- [On the Emergence of Cross-Task Linearity in the Pretraining-Finetuning Paradigm](https://mlsanity.com/paper/2402.03660/) (2024)
- [PCP-MAE: Learning to Predict Centers for Point Masked Autoencoders](https://mlsanity.com/paper/2408.08753/) (2024)
- [Understanding Visual Feature Reliance through the Lens of Complexity](https://mlsanity.com/paper/2407.06076/) (2024)
- [From Task-Specific Models to Unified Systems: A Review of Model Merging Approaches](https://mlsanity.com/paper/2503.08998/) (2025)
- [Forget the Data and Fine-Tuning! Just Fold the Network to Compress](https://mlsanity.com/paper/2502.10216/) (2025)
- [Cut Less, Fold More: Model Compression through the Lens of Projection Geometry](https://mlsanity.com/paper/2602.18116/) (2026)
- [BALF: Budgeted Activation-Aware Low-Rank Factorization for Fine-Tuning-Free Model Compression](https://mlsanity.com/paper/2509.25136/) (2025)
- [GRAIL: Post-hoc Compensation by Linear Reconstruction for Compressed Networks](https://mlsanity.com/paper/2602.23795/) (2026)
- [From Intent to Evidence: A Categorical Approach for Structural Evaluation of Deep Research Agents](https://mlsanity.com/paper/2603.25342/) (2026)
- [Entropy Distribution as a Fingerprint for Hallucinations in Generative Models](https://mlsanity.com/paper/2605.28264/) (2026)

## Abstract (arXiv)

The behavior of neural networks still remains opaque, and a recently widely noted phenomenon is that networks often achieve similar performance when initialized with different random parameters. This phenomenon has attracted significant attention in measuring the similarity between features learned by distinct networks. However, feature similarity could be vague in describing the same feature since equivalent features hardly exist. In this paper, we expand the concept of equivalent feature and provide the definition of what we call functionally equivalent features. These features produce equivalent output under certain transformations. Using this definition, we aim to derive a more intrinsic metric for the so-called feature complexity regarding the redundancy of features learned by a neural network at each layer. We offer a formal interpretation of our approach through the lens of category theory, a well-developed area in mathematics. To quantify the feature complexity, we further propose an efficient algorithm named Iterative Feature Merging. Our experimental results validate our ideas and theories from various perspectives. We empirically demonstrate that the functionally equivalence widely exists among different features learned by the same neural network and we could reduce the number of parameters of the network without affecting the performance.The IFM shows great potential as a data-agnostic model prune method. We have also drawn several interesting empirical findings regarding the defined feature complexity.