Effective feature representation is key to the predictive performance of any
algorithm. This paper introduces a meta-procedure, called Non-Euclidean
Upgrading (NEU), which learns feature maps that are expressive enough to embed
the universal approximation property (UAP) into most model classes while only
outputting feature maps that preserve any model class's UAP. We show that NEU
can learn any feature map with these two properties if that feature map is
asymptotically deformable into the identity. We also find that the
feature-representations learned by NEU are always submanifolds of the feature
space. NEU's properties are derived from a new deep neural model that is
universal amongst all orientation-preserving homeomorphisms on the input space.
We derive qualitative and quantitative approximation guarantees for this
architecture. We quantify the number of parameters required for this new
architecture to memorize any set of input-output pairs while simultaneously
fixing every point of the input space lying outside some compact set, and we
quantify the size of this set as a function of our model's depth. Moreover, we
show that no deep feed-forward network with commonly used activation function
has all these properties. NEU's performance is evaluated against competing
machine learning methods on various regression and dimension reduction tasks
both with financial and simulated data.

本文介绍了一种名为 Non-Euclidean Upgrading 的元程序，它通过学习特征映射来嵌入大多数模型类的全局逼近性质，并且学习得到的特征表达式总是特征空间的亚流形。

NEU：一种通用的 UAP 不变特征表征元算法

NEU: A Meta-Algorithm for Universal UAP-Invariant Feature Representation

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。