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.

利用对得分映射的固定点 insight，通过利用对应的狄利克雷能量进行正则化将其前提条件确认为对潜在 manifold 学习的拓扑维度进行回归是可能的，然后引入了一种使用对抗性攻击来测量学习的 manifold 的拓扑维度的新方法，从而产生对学习的 manifold 的有用解释。

用对抗方法估计拓扑维度的谐波评分图

Adversarial Estimation of Topological Dimension with Harmonic Score Maps

The existing approaches to intrinsic dimension estimation usually are not
reliable when the data are nonlinearly embedded in the high dimensional space.
In this work, we show that the explicit accounting to geometric properties of
unknown support leads to the polynomial correction to the standard maximum
likelihood estimate of intrinsic dimension for flat manifolds. The proposed
algorithm (GeoMLE) realizes the correction by regression of standard MLEs based
on distances to nearest neighbors for different sizes of neighborhoods.
Moreover, the proposed approach also efficiently handles the case of nonuniform
sampling of the manifold. We perform numerous experiments on different
synthetic and real-world datasets. The results show that our algorithm achieves
state-of-the-art performance, while also being computationally efficient and
robust to noise in the data.

本论文提出了一种名为 GeoMLE 的算法，通过回归标准 MLEs 实现对平坦流形的内在维度的显式计算，能够有效地处理非均匀采样的流形，并在不同的合成和真实数据集上实现了最先进的性能。