Advanced representation learning techniques require reliable and general
evaluation methods. Recently, several algorithms based on the common idea of
geometric and topological analysis of a manifold approximated from the learned
data representations have been proposed. In this work, we introduce Delaunay
Component Analysis (DCA) - an evaluation algorithm which approximates the data
manifold using a more suitable neighbourhood graph called Delaunay graph. This
provides a reliable manifold estimation even for challenging geometric
arrangements of representations such as clusters with varying shape and density
as well as outliers, which is where existing methods often fail. Furthermore,
we exploit the nature of Delaunay graphs and introduce a framework for
assessing the quality of individual novel data representations. We
experimentally validate the proposed DCA method on representations obtained
from neural networks trained with contrastive objective, supervised and
generative models, and demonstrate various use cases of our extended single
point evaluation framework.

本文提出了一种基于 Delaunay 图的 Delaunay Component Analysis (DCA) 评估算法，它可靠地估计了数据流形，即使对于表示具有具有变化形状和密度的簇以及离群值的情况，这是现有方法常常失败的地方，同时还引入了用于评估个体新数据表示质量的框架，并在不同的单点评估框架中使用神经网络训练的表示进行了实验验证。

用 Delaunay 分量分析评估数据表征

Delaunay Component Analysis for Evaluation of Data Representations

Various problems in manifold estimation make use of a quantity called the
reach, denoted by $\tau\_M$, which is a measure of the regularity of the
manifold. This paper is the first investigation into the problem of how to
estimate the reach. First, we study the geometry of the reach through an
approximation perspective. We derive new geometric results on the reach for
submanifolds without boundary. An estimator $\hat{\tau}$ of $\tau\_{M}$ is
proposed in a framework where tangent spaces are known, and bounds assessing
its efficiency are derived. In the case of i.i.d. random point cloud
$\mathbb{X}\_{n}$, $\hat{\tau}(\mathbb{X}\_{n})$ is showed to achieve uniform
expected loss bounds over a $\mathcal{C}^3$-like model. Finally, we obtain
upper and lower bounds on the minimax rate for estimating the reach.

该论文是关于如何估算流形的正则程度的研究。提出了一个估计器来估计流形的 reach，给出了达到统一的期望损失界限的证明，并给出了针对 reach 估计的最小化率的上下界。

估计流形的范围

Estimating the Reach of a Manifold

Topological Data Analysis (TDA) can broadly be described as a collection of
data analysis methods that find structure in data. This includes: clustering,
manifold estimation, nonlinear dimension reduction, mode estimation, ridge
estimation and persistent homology. This paper reviews some of these methods.

本文综述了拓扑数据分析中用于数据结构发现的聚类、流形估计、非线性降维、模式估计、脊估计和持久同调等方法。

拓扑数据分析

Topological Data Analysis

We study the problem of estimating, in the sense of optimal transport
metrics, a measure which is assumed supported on a manifold embedded in a
Hilbert space. By establishing a precise connection between optimal transport
metrics, optimal quantization, and learning theory, we derive new probabilistic
bounds for the performance of a classic algorithm in unsupervised learning
(k-means), when used to produce a probability measure derived from the data. In
the course of the analysis, we arrive at new lower bounds, as well as
probabilistic upper bounds on the convergence rate of the empirical law of
large numbers, which, unlike existing bounds, are applicable to a wide class of
measures.

通过优化传输度量，在嵌入 Hilbert 空间的流形上估计一种衡量方法，并将量化优化和学习理论联系起来，为无监督学习中经典算法（k-means）的性能提供新的概率界限。在分析的过程中，我们得出了新的下界和概率上界，这些上下界适用于广泛的测度范围。