We introduce a novel method for acquiring boundary representations (B-Reps)
of 3D CAD models which involves a two-step process: it first applies a spatial
partitioning, referred to as the ``split``, followed by a ``fit`` operation to
derive a single primitive within each partition. Specifically, our partitioning
aims to produce the classical Voronoi diagram of the set of ground-truth (GT)
B-Rep primitives. In contrast to prior B-Rep constructions which were
bottom-up, either via direct primitive fitting or point clustering, our
Split-and-Fit approach is top-down and structure-aware, since a Voronoi
partition explicitly reveals both the number of and the connections between the
primitives. We design a neural network to predict the Voronoi diagram from an
input point cloud or distance field via a binary classification. We show that
our network, coined NVD-Net for neural Voronoi diagrams, can effectively learn
Voronoi partitions for CAD models from training data and exhibits superior
generalization capabilities. Extensive experiments and evaluation demonstrate
that the resulting B-Reps, consisting of parametric surfaces, curves, and
vertices, are more plausible than those obtained by existing alternatives, with
significant improvements in reconstruction quality. Code will be released on
this https URL

我们介绍了一种新的方法来获取 3D CAD 模型的边界表示（B-Reps），这涉及两个步骤：首先应用空间划分，称为 ``拆分 ''，然后通过`` 拟合 '' 操作在每个划分内导出一个单一的基元。我们的划分旨在产生一组地面真值（GT）B-Rep 基元的经典的 Voronoi 图；我们的方法采用自上而下的结构感知方式，能够明确显示基元的数量和连接关系。我们设计了一个神经网络，通过二进制分类从输入点云或距离场预测 Voronoi 图。我们展示了我们的网络 NVD-Net（神经 Voronoi 图）可以有效地从训练数据学习 CAD 模型的 Voronoi 划分，并具有更好的泛化能力。广泛的实验和评估表明，与现有的替代方法相比，所得到的由参数曲面，曲线和顶点组成的 B-Reps 更加合理，重建质量有了显著的提高。代码将在此 https URL 上发布。

分割与匹配：通过结构感知的 Voronoi 分区学习 B-Reps

Split-and-Fit: Learning B-Reps via Structure-Aware Voronoi Partitioning

Exploring whether Enriched Category Theory could provide the foundation of an
alternative approach to Machine Learning. This paper is the first to construct
and motivate a Machine Learning algorithm solely with Enriched Category Theory.
In order to supplement evidence that Category Theory can be used to motivate
robust and explainable algorithms, it is shown that a series of reasonable
assumptions about a dataset lead to the construction of the Nearest Neighbours
Algorithm. In particular, as an extension of the original dataset using
profunctors in the category of Lawvere metric spaces. This leads to a
definition of an Enriched Nearest Neighbours Algorithm, which consequently also
produces an enriched form of the Voronoi diagram. This paper is intended to be
accessible without any knowledge of Category Theory

使用丰富范畴论为基础构建了一种新型机器学习算法，该算法仅使用丰富范畴论；通过一系列关于数据集的合理假设构建了最近邻算法，其中利用了范畴中的 Lawvere 度量空间的协函子对扩展原始数据集；从而定义了一种丰富最近邻算法，并产生了一个丰富的 Voronoi 图。本文旨在让无范畴论知识的读者也能理解。

利用丰富的范畴论构建最近邻分类算法

Using Enriched Category Theory to Construct the Nearest Neighbour  Classification Algorithm

Few-shot learning (FSL) is the process of rapid generalization from abundant
base samples to inadequate novel samples. Despite extensive research in recent
years, FSL is still not yet able to generate satisfactory solutions for a wide
range of real-world applications. To confront this challenge, we study the FSL
problem from a geometric point of view in this paper. One observation is that
the widely embraced ProtoNet model is essentially a Voronoi Diagram (VD) in the
feature space. We retrofit it by making use of a recent advance in
computational geometry called Cluster-induced Voronoi Diagram (CIVD). Starting
from the simplest nearest neighbor model, CIVD gradually incorporates
cluster-to-point and then cluster-to-cluster relationships for space
subdivision, which is used to improve the accuracy and robustness at multiple
stages of FSL. Specifically, we use CIVD (1) to integrate parametric and
nonparametric few-shot classifiers; (2) to combine feature representation and
surrogate representation; (3) and to leverage feature-level,
transformation-level, and geometry-level heterogeneities for a better ensemble.
Our CIVD-based workflow enables us to achieve new state-of-the-art results on
mini-ImageNet, CUB, and tiered-ImagenNet datasets, with ${\sim}2\%{-}5\%$
improvements upon the next best. To summarize, CIVD provides a mathematically
elegant and geometrically interpretable framework that compensates for extreme
data insufficiency, prevents overfitting, and allows for fast geometric
ensemble for thousands of individual VD. These together make FSL stronger.

本文从几何角度研究了少样本学习的问题，并提出了一个基于 Cluster-induced Voronoi Diagram 的方法，用于集成参数和非参数，进行特征表示和代理表示的组合，以及利用几何层次的异质性进行改进。该方法在 mini-ImageNet、CUB 和 tiered-ImagenNet 数据集上创造了新的最优结果，比第二好的结果提高了大约 2%-5%。