# Geometric and Topological Inference for Deep Representations of Complex Networks

arXiv: https://arxiv.org/abs/2203.05488 · HTML: https://mlsanity.com/paper/2203.05488/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 36 references that are in the map.
- First posted on arXiv 10 Mar 2022.

## Builds on (most cited first, 9 of 9)

- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [Insights on representational similarity in neural networks with canonical correlation](https://mlsanity.com/paper/1806.05759/) (2018)
- [Universality and individuality in neural dynamics across large populations of recurrent networks](https://mlsanity.com/paper/1907.08549/) (2019)
- [Bridging the Gaps Between Residual Learning, Recurrent Neural Networks and Visual Cortex](https://mlsanity.com/paper/1604.03640/) (2016)
- [Demixed principal component analysis of population activity in higher cortical areas reveals independent representation of task parameters](https://mlsanity.com/paper/1410.6031/) (2014)
- [Topological analysis of the connectome of digital reconstructions of neural microcircuits](https://mlsanity.com/paper/1601.01580/) (2016)
- [Visualizing Representational Dynamics with Multidimensional Scaling Alignment](https://mlsanity.com/paper/1906.09264/) (2019)
- [Regularity Normalization: Neuroscience-Inspired Unsupervised Attention across Neural Network Layers](https://mlsanity.com/paper/1902.10658/) (2019)

## Built on by (most cited first, 4 of 4)

- [Similarity of Neural Network Models: A Survey of Functional and Representational Measures](https://mlsanity.com/paper/2305.06329/) (2023)
- [Universal Neurons in GPT2 Language Models](https://mlsanity.com/paper/2401.12181/) (2024)
- [The Topology and Geometry of Neural Representations](https://mlsanity.com/paper/2309.11028/) (2023)
- [Topological Data Analysis in Time Series: Temporal Filtration and Application to Single-Cell Genomics](https://mlsanity.com/paper/2204.14048/) (2022)

## Abstract (arXiv)

Understanding the deep representations of complex networks is an important step of building interpretable and trustworthy machine learning applications in the age of internet. Global surrogate models that approximate the predictions of a black box model (e.g. an artificial or biological neural net) are usually used to provide valuable theoretical insights for the model interpretability. In order to evaluate how well a surrogate model can account for the representation in another model, we need to develop inference methods for model comparison. Previous studies have compared models and brains in terms of their representational geometries (characterized by the matrix of distances between representations of the input patterns in a model layer or cortical area). In this study, we propose to explore these summary statistical descriptions of representations in models and brains as part of a broader class of statistics that emphasize the topology as well as the geometry of representations. The topological summary statistics build on topological data analysis (TDA) and other graph-based methods. We evaluate these statistics in terms of the sensitivity and specificity that they afford when used for model selection, with the goal to relate different neural network models to each other and to make inferences about the computational mechanism that might best account for a black box representation. These new methods enable brain and computer scientists to visualize the dynamic representational transformations learned by brains and models, and to perform model-comparative statistical inference.