Metric-Aware PCA as a Linear Instance of Geometric Deep Learning
This paper introduces Metric-Aware Principal Component Analysis (MAPCA), which parameterizes PCA with a positive-definite metric matrix and positions it within the geometric deep learning framework. MAPCA interprets the metric as a geometric prior, its solutions are equivariant under the orthogonal group preserving the metric, and its spectrum is invariant. A uniqueness theorem characterizes Invariant PCA (IPCA) as the unique linear data-derived metric in the MAPCA family that is equivariant under arbitrary diagonal rescaling. The paper also discusses extensions to kernel PCA, spectral graph methods, and deep MAPCA.
Article intelligence
Key points
- MAPCA parameterizes PCA with a positive-definite metric matrix, linking geometric deep learning symmetry and equivariance concepts.
- A uniqueness theorem shows that IPCA is the unique linear data-derived metric in the MAPCA family equivariant under diagonal rescaling.
- The paper establishes a precise dictionary between MAPCA and geometric deep learning across six axes.
- Extensions include kernel PCA, spectral graph methods, and deep MAPCA, demonstrating the framework's generality.
Why it matters
This matters because MAPCA parameterizes PCA with a positive-definite metric matrix, linking geometric deep learning symmetry and equivariance concepts.
Technical impact
May affect research directions, evaluation methods, open-source reproduction, and productization paths.
[2605.27456] Metric-Aware PCA as a Linear Instance of Geometric Deep Learning
[Submitted on 25 May 2026]
Title:Metric-Aware PCA as a Linear Instance of Geometric Deep Learning
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Abstract: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
Subjects:
Machine Learning (cs.LG)
Cite as: arXiv:2605.27456 [cs.LG]
(or arXiv:2605.27456v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2605.27456
arXiv-issued DOI via DataCite (pending registration)
Submission history
From: Michael Leznik Dr. [view email] [v1] Mon, 25 May 2026 16:07:24 UTC (18 KB)
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