# Adversarial Estimation of Topological Dimension with Harmonic Score Maps

arXiv: https://arxiv.org/abs/2312.06869 · HTML: https://mlsanity.com/paper/2312.06869/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 33 references that are in the map.
- First posted on arXiv 11 Dec 2023.

## Builds on (most cited first, 21 of 21)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Towards Evaluating the Robustness of Neural Networks](https://mlsanity.com/paper/1608.04644/) (2016)
- [Certified Adversarial Robustness via Randomized Smoothing](https://mlsanity.com/paper/1902.02918/) (2019)
- [DiGress: Discrete Denoising diffusion for graph generation](https://mlsanity.com/paper/2209.14734/) (2022)
- [Diffusion Models for Adversarial Purification](https://mlsanity.com/paper/2205.07460/) (2022)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Guided Diffusion Model for Adversarial Purification](https://mlsanity.com/paper/2205.14969/) (2022)
- [Scikit-dimension: a Python package for intrinsic dimension estimation](https://mlsanity.com/paper/2109.02596/) (2021)
- [Statistical Efficiency of Score Matching: The View from Isoperimetry](https://mlsanity.com/paper/2210.00726/) (2022)
- [Why neural networks find simple solutions: the many regularizers of geometric complexity](https://mlsanity.com/paper/2209.13083/) (2022)
- [LIDL: Local Intrinsic Dimension Estimation Using Approximate Likelihood](https://mlsanity.com/paper/2206.14882/) (2022)
- [NashAE: Disentangling Representations through Adversarial Covariance Minimization](https://mlsanity.com/paper/2209.10677/) (2022)
- [Adversarial Robustness of Flow-Based Generative Models](https://mlsanity.com/paper/1911.08654/) (2019)
- [Disentangling Learning Representations with Density Estimation](https://mlsanity.com/paper/2302.04362/) (2023)

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

- [A Connection Between Score Matching and Local Intrinsic Dimension](https://mlsanity.com/paper/2510.12975/) (2025)
- [Do Counterfactual Examples Complicate Adversarial Training?](https://mlsanity.com/paper/2404.10588/) (2024)
- [A Wiener Process Perspective on Local Intrinsic Dimension Estimation Methods](https://mlsanity.com/paper/2406.17125/) (2024)
- [Saddle-Free Guidance: Improved On-Manifold Sampling without Labels or Additional Training](https://mlsanity.com/paper/2511.21863/) (2025)

## Abstract (arXiv)

Quantification of the number of variables needed to locally explain complex data is often the first step to better understanding it. Existing techniques from intrinsic dimension estimation leverage statistical models to glean this information from samples within a neighborhood. However, existing methods often rely on well-picked hyperparameters and ample data as manifold dimension and curvature increases. Leveraging insight into the fixed point of the score matching objective as the score map is regularized by its Dirichlet energy, we show that it is possible to retrieve the topological dimension of the manifold learned by the score map. We then introduce a novel method to measure the learned manifold's topological dimension (i.e., local intrinsic dimension) using adversarial attacks, thereby generating useful interpretations of the learned manifold.