Metric-Aware PCA as a Linear Instance of Geometric Deep Learning
For researchers in geometric deep learning and dimensionality reduction, this work provides a theoretical bridge between linear methods and equivariant networks, but is primarily a conceptual positioning rather than a practical advance.
This paper introduces Metric-Aware PCA (MAPCA), a linear method parameterized by a positive-definite metric matrix, and positions it within geometric deep learning by showing its equivariance and invariance properties. It establishes a precise dictionary between MAPCA and geometric deep learning across six axes, and proves a uniqueness theorem characterizing Invariant PCA as the unique linear data-derived metric equivariant under diagonal rescaling.
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