PCA-Based Interpretable Knowledge Representation and Analysis of Geometric Design Parameters

Geometric deep learning organises neural architectures around the symmetries of their data domain, with the choice of symmetry group serving as a geometric prior that determines what representations can be learned. Metric-Aware Principal Component Analysis (MAPCA) parameterises principal component analysis by a positive-definite metric matrix, with a canonical subfamily interpolating between standard PCA and output whitening and a diagonal-metric point recovering Invariant PCA (IPCA). This paper positions MAPCA within the geometric deep learning framework. The metric is read as the geometric prior; the orthogonal group preserving it is the symmetry group it induces; MAPCA solutions are equivariant under this group with the resulting spectrum invariant; and MAPCA's defining constraint is the linear analogue of the Schur-type weight constraints used in equivariant networks. Across six axes - domain, symmetry group, equivariance, invariance, architectural primitive, and geometric prior - we construct a precise dictionary between MAPCA and geometric deep learning. The technical anchor is a uniqueness theorem characterising IPCA as the unique linear data-derived metric in the MAPCA family that is equivariant under arbitrary diagonal rescaling and projects onto the fixed-point set of the action, equivalent under normalisation to the variance-maximisation criterion in its precise form. The paper closes with three bridges: kernel PCA as the nonlinear extension, spectral graph methods as MAPCA on graphs, and a deep MAPCA construction extending the positioning into deep equivariant networks

Dimensionality reduction is essential in simulation-based shape design, where high-dimensional parameterizations hinder optimization, surrogate modeling, and systematic design-space exploration. Parametric Model Embedding (PME) addresses this issue by constructing reduced variables from geometric information while preserving an explicit backmapping to the original design parameters. However, PME is intrinsically linear and may become inefficient when the sampled design space is governed by nonlinear geometric variability. This paper introduces a nonlinear extension of PME, denoted NLPME. The proposed framework preserves the defining principle of PME -- geometry-driven latent variables and parameter-mediated reconstruction -- while replacing the linear reduced subspace with a nonlinear latent representation. Geometry is not reconstructed directly from the latent variables; instead, the latent representation is decoded into admissible design parameters, and the corresponding geometry is recovered through a forward parametric map. The method is assessed on a bio-inspired autonomous underwater glider with a 32-dimensional parametric shape description and a CAD-based geometry-generation process. NLPME reaches a 5% reconstruction-error threshold with $N=5$ latent variables, compared with $N=8$ for linear PME, and a 1% threshold with $N=9$, compared with $N=15$ for PME. Comparison with a deep autoencoder shows that most of the nonlinear compression gain can be retained while preserving an explicit backmapping to the original design variables. The results establish NLPME as a compact, admissible, and engineering-compatible nonlinear reduced representation for parametric shape design spaces.

In this paper, we propose SHOPCA (Shape Operator-based Principal Component Analysis), a novel method for unsupervised metric learning and dimensionality reduction that incorporates differential geometric information into the covariance structure of classical PCA. SHOPCA regularizes the global covariance matrix using the mean shape operator, defined as the average of the absolute local shape operators estimated from the data manifold, steering principal components toward directions of both maximum variance and informative curvature. A single trace-normalized mixing coefficient $α$ controls the regularization, recovering standard PCA at $α= 0$ and a curvature-driven embedding as $α\to \infty$. We further introduce a fully unsupervised criterion for selecting $α$ based on the spectral eigengap of the regularized covariance matrix, maximizing the relative separation between the top-$d$ and remaining eigenvalues without using class labels. We evaluate SHOPCA on more than 50 real-world benchmark datasets, comparing it with PCA, ISOMAP, and UMAP using Adjusted Rand Index (ARI), Normalized Mutual Information (NMI), Fowlkes-Mallows index (FM), and V-measure. Results show that SHOPCA consistently improves clustering quality over PCA across a broad range of datasets and surpasses UMAP on small-sample settings, where iterative neighborhood-based manifold estimation can degrade. SHOPCA is computationally tractable, parameter-efficient, and applicable to domains requiring fully unsupervised, geometry-aware dimensionality reduction.