We introduce an algorithm, Cayley transform ellipsoid fitting (CTEF), that
uses the Cayley transform to fit ellipsoids to noisy data in any dimension.
Unlike many ellipsoid fitting methods, CTEF is ellipsoid specific -- meaning it
always returns elliptic solutions -- and can fit arbitrary ellipsoids. It also
outperforms other fitting methods when data are not uniformly distributed over
the surface of an ellipsoid. Inspired by calls for interpretable and
reproducible methods in machine learning, we apply CTEF to dimension reduction,
data visualization, and clustering. Since CTEF captures global curvature, it is
able to extract nonlinear features in data that other methods fail to identify.
This is illustrated in the context of dimension reduction on human cell cycle
data, and in the context of clustering on classical toy examples. In the latter
case, CTEF outperforms 10 popular clustering algorithms.

本文提出一种基于 Cayley 变换的椭球拟合算法，可以适应各种维度和任意的椭球形状，且比其他方法表现更好，该算法应用于降维、数据可视化和数据聚类，并能识别非线性特征。

Cayley 变换下的椭球拟合

Ellipsoid fitting with the Cayley transform

In random sample consensus (RANSAC), the problem of ellipsoid fitting can be
formulated as a problem of minimization of point-to-model distance, which is
realized by maximizing model score. Hence, the performance of ellipsoid fitting
is affected by distance metric. In this paper, we proposed a novel distance
metric called the axial distance, which is converted from the algebraic
distance by introducing a scaling factor to solve nongeometric problems of the
algebraic distance. There is complementarity between the axial distance and
Sampson distance because their combination is a stricter metric when
calculating the model score of sample consensus and the weight of the weighted
least squares (WLS) fitting. Subsequently, a novel sample-consensus-based
ellipsoid fitting method is proposed by using the combination between the axial
distance and Sampson distance (CAS). We compare the proposed method with
several representative fitting methods through experiments on synthetic and
real datasets. The results show that the proposed method has a higher
robustness against outliers, consistently high accuracy, and a speed close to
that of the method based on sample consensus.

本文提出了一种基于样本一致性的新型椭球拟合方法，使用轴向距离和 Sampson 距离的组合进行计算，在合成和实际数据集上的实验结果表明，所提出的方法具有更高的鲁棒性，一致的精度和接近于基于样本一致性方法的速度。