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

arXiv: https://arxiv.org/abs/2605.27456 · HTML: https://mlsanity.com/paper/2605.27456/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 9 references that are in the map.
- First posted on arXiv 25 May 2026.

## Builds on (most cited first, 4 of 4)

- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Whitening for Self-Supervised Representation Learning](https://mlsanity.com/paper/2007.06346/) (2020)
- [Metric-Aware Principal Component Analysis (MAPCA):A Unified Framework for Scale-Invariant Representation Learning](https://mlsanity.com/paper/2604.14249/) (2026)

## Abstract (arXiv)

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