Orthogonality and Dimensionality in Airline Cluster Analysis using PCA and Kernel PCA

We address shortcomings of principal component analysis (PCA) for visualizing high-dimensional data lying on a nonlinear low-dimensional manifold via two-dimensional scatterplots, focusing on a fossil teeth dataset from the early mammalian insectivore Kuehneotherium. While the PCA scatterplot reported by Jolliffe and Cadima (Philosophical Transactions of the Royal Society A, 2016) shows clustering in the region where PC2 < 0, our analysis based on t-SNE and persistent homology (PH) reveals a ring-like structure with no evident clustering and intrinsic dimensionality equal to one. We further propose a generative probabilistic-geometric model in which the data are sampled uniformly from a unit circle. Under this model, pairwise cosine distances follow an arcsine distribution, in qualitative agreement with the observed U-shaped distribution, thereby independently supporting the analysis based on t-SNE and persistent homology.

K-plane clustering (KPC), hyperplane clustering, and mixture regression all essentially fall within the same class of problems. This problem can be conceptualized as clustering in relatively high-dimensional K subspaces or K linear manifolds. Traditional KPC or fuzzy KPC models demonstrate a pronounced susceptibility to outliers, as they presuppose that the projection distance between data points and the plane normal vector adheres to the L2 distance. Meanwhile, the assumption of infinitely extending clusters adversely affects clustering performance. To solve these problems, this paper proposed a new robust fuzzy local k-plane clustering (RFLkPC) method that combines the mixture distance of hinge loss and L1 norm. The RFLkPC model assumes that each plane cluster is bounded to a finite area, which can flexibly and robustly handle plane clustering tasks with outliers or not. The corresponding model and optimization algorithms of RFLkPC were provided. Compared to other related models on this topic, a large number of experiments verify the efficiency of RFLkPC on simulated data and real data. The source code for the proposed RFLkPC method is publicly available at https://github.com/xuelin-xie/RFLkPC.

Principal component analysis (PCA) is one of the most widely used unsupervised dimension reduction techniques. We study PCA for data from multiple related domains. Since principal components generally differ across domains, one way to obtain a shared low-rank embedding is to perform PCA on the pooled data. However, this approach can focus on spurious directions that exhibit high variation in only a few domains. To find a robust embedding that still explains most variance in unseen but similar domains, we propose instead to focus on shared directions of variation. To this end, we introduce Anchor PCA which trades off overall explained variance with agreement between the shared and domain-specific low-rank embeddings. Anchor PCA amounts to PCA on a modified target matrix and thus can be solved efficiently. Moreover, we show that Anchor PCA recovers a maximal invariant subspace and admits a minimax reconstruction interpretation under bounded domain-specific covariance inflations. On simulated and real-world gas sensor data with temporal drift, we demonstrate, respectively, that Anchor PCA recovers the maximally invariant subspace and yields embeddings that explain more variance on unseen domains than the pooling baseline and a worst-case alternative. Taken together, these findings establish Anchor PCA as a promising approach to robust unsupervised dimension reduction from multi-domain data.