Metric-Aware PCA as a Linear Instance of Geometric Deep Learning

A new theoretical paper positions Metric-Aware Principal Component Analysis (MAPCA) as a linear instance of geometric deep learning, constructing a precise dictionary between the two frameworks across six axes and establishing a uniqueness theorem for Invariant PCA (IPCA) [1][2]. The work, submitted to arXiv on 25 May 2026, parameterises principal component analysis by a positive-definite metric matrix [1][2]. Within this family, a canonical subfamily interpolates between standard PCA and output whitening, while a diagonal-metric point recovers IPCA [2]. The authors read the metric as the geometric prior and the orthogonal group preserving it as the induced symmetry group, showing that MAPCA solutions are equivariant under this group and that the resulting spectrum remains invariant [1][2]. The defining constraint of MAPCA is presented as the linear analogue of the Schur-type weight constraints used in equivariant networks [1][2]. 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 [2]. Neural networks themselves consist of connected units called artificial neurons, which process signals through weighted connections that adjust during training; deep neural networks, defined as having at least two hidden layers, are capable of learning sophisticated hierarchical representations [3]. The MAPCA framework translates these principles into a linear setting. The paper's 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 [1][2]. Under normalisation, this is equivalent to the variance-maximisation criterion in its precise form [2]. The work closes with three bridges: kernel PCA as the nonlinear extension of MAPCA, spectral graph methods as MAPCA on graphs, and a deep MAPCA construction that extends the positioning into deep equivariant networks [1][2]. While the paper does not introduce new datasets, high-quality training datasets remain integral to machine learning research, with major advances often resulting from improvements in algorithms, hardware, and data availability [5].