Platonic Projection Structures: Operator-Induced Observability in Representation Learning

• URL: http://arxiv.org/abs/2607.05175v1
• Date: Mon, 06 Jul 2026 14:57:26 GMT
• Title: Platonic Projection Structures: Operator-Induced Observability in Representation Learning
• Abstract summary: We characterize observability in representation learning through Platonic Projection Structures (PPS)<n>PPS models observation through a self-adjoint positive semidefinite operator acting on a latent representation space.<n>We show that quantum measurement and representation inference under linear observation models share this operator-theoretic structure.
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• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We characterize observability in representation learning through Platonic Projection Structures (PPS), an operator-theoretic framework for analyzing representation accessibility under partial observation. Rather than treating observable outputs as direct reflections of latent representations, PPS models observation through a self-adjoint positive semidefinite operator acting on a latent representation space. A system is represented as a triple $(H, Π, O)$, where $H$ is a latent representation space, $Π\succeq 0$ is an observation operator, and $O(v)=\langle v,Πv\rangle$ defines an induced scalar observable. Observability is characterized by the quotient geometry $H/\ker(Π)$, representing equivalence classes of latent states indistinguishable under observation. We show that quantum measurement and representation inference under linear observation models share this operator-theoretic structure while differing in the algebraic properties of their observation operators; the correspondence is structural rather than physical. Representation transfer and knowledge distillation can likewise be interpreted as approximate preservation of observable geometry through $ΦΠ_T \approx Π_S Φ$. PPS also reveals a structural limitation of output-based interpretability: latent components in $\ker(Π)$ are inaccessible from induced observables, imposing intrinsic constraints on attribution and explanation methods. Controlled empirical validations demonstrate kernel-invariant observability, projection-induced attribution gaps, and rank-controlled observable geometry in latent representation spaces. PPS thus provides an explicit characterization of observability through operator-induced quotient geometry and a unified perspective on representation accessibility, interpretability, and projection-mediated inference.

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