Recent studies propose enhancing machine learning models by aligning the
geometric characteristics of the latent space with the underlying data
structure. Instead of relying solely on Euclidean space, researchers have
suggested using hyperbolic and spherical spaces with constant curvature, or
their combinations (known as product manifolds), to improve model performance.
However, there exists no principled technique to determine the best latent
product manifold signature, which refers to the choice and dimensionality of
manifold components. To address this, we introduce a novel notion of distance
between candidate latent geometries using the Gromov-Hausdorff distance from
metric geometry. We propose using a graph search space that uses the estimated
Gromov-Hausdorff distances to search for the optimal latent geometry. In this
work we focus on providing a description of an algorithm to compute the
Gromov-Hausdorff distance between model spaces and its computational
implementation.

为了改善模型性能，最近的研究提出通过使潜空间的几何特征与基础数据结构对齐来提升机器学习模型。本文引入了一种使用 Gromov-Hausdorff 距离计算候选潜空间几何之间距离的新概念，提出了使用估计的 Gromov-Hausdorff 距离来搜索最佳潜空间几何的图搜索空间算法。本文描述了计算模型空间之间 Gromov-Hausdorff 距离的算法及其计算实现。

比较模型空间的产品流形的 Gromov-Hausdorff 距离

Gromov-Hausdorff Distances for Comparing Product Manifolds of Model  Spaces

This paper presents connections between Gromov's work on isoperimetry of
waists and Milman's work on the $M$-ellipsoid of a convex body. It is proven
that any convex body $K \subseteq \mathbb{R}^n$ has a linear image $\tilde{K}
\subseteq \mathbb{R}^n$ of volume one satisfying the following waist
inequality: Any continuous map $f:\tilde{K} \rightarrow \mathbb{R}^{\ell}$ has
a fiber $f^{-1}(t)$ whose $(n-\ell)$-dimensional volume is at least
$c^{n-\ell}$, where $c > 0$ is a universal constant. In the specific case where
$K = [0,1]^n$ it is shown that one may take $\tilde{K} = K$ and $c = 1$,
confirming a conjecture by Guth. We furthermore exhibit relations between waist
inequalities and various geometric characteristics of the convex body $K$.

本文研究了 Gromov 的等周腰定理和 Milman 的 M - 椭球对凸体的作用，证明了任何凸体都有体积为 1 的线性映射，并且展示了等周腰不等式与凸体各种几何特征之间的联系。

凸几何与腰不等式

Convex geometry and waist inequalities

Empirically validating new 3D-printing related algorithms and implementations
requires testing data representative of inputs encountered \emph{in the wild}.
An ideal benchmarking dataset should not only draw from the same distribution
of shapes people print in terms of class (e.g., toys, mechanisms, jewelry),
representation type (e.g., triangle soup meshes) and complexity (e.g., number
of facets), but should also capture problems and artifacts endemic to 3D
printing models (e.g., self-intersections, non-manifoldness). We observe that
the contextual and geometric characteristics of 3D printing models differ
significantly from those used for computer graphics applications, not to
mention standard models (e.g., Stanford bunny, Armadillo, Fertility). We
present a new dataset of 10,000 models collected from an online 3D printing
model-sharing database. Via analysis of both geometric (e.g., triangle aspect
ratios, manifoldness) and contextual (e.g., licenses, tags, classes)
characteristics, we demonstrate that this dataset represents a more concise
summary of real-world models used for 3D printing compared to existing
datasets. To facilitate future research endeavors, we also present an online
query interface to select subsets of the dataset according to project-specific
characteristics. The complete dataset and per-model statistical data are freely
available to the public.

本文介绍了一种更加贴近现实世界中 3D 打印模型的新的基准测试数据集，并提供了在线查询接口选择数据集子集以便未来的研究。