aiqbench analysis
This paper introduces a novel operator-theoretic framework, Platonic Projection Structures (PPS), to rigorously characterize observability limits in representation learning under partial observation. By modeling observation as a self-adjoint operator, the work unifies concepts from quantum measurement theory with classical machine learning inference, revealing intrinsic constraints on interpretability where latent components remain inaccessible to output-based attribution methods. The study provides explicit mathematical characterizations for knowledge distillation and transfer learning through the preservation of observable geometry.
Platonic Projection Structures: Operator-Induced Observability in Representation Learning
arXiv:2607.05175v2 Announce Type: replace 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 $(H, \Pi, O)$, where $H$ is a latent representation space, $\Pi \succeq 0$ is an observation operator, and $O(v)=\langle v,\Pi v\rangle$ defines an induced scalar observable. Observability is characterized by the quotient geometry $H/\ker(\Pi)$, 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 $\Phi \Pi_T \approx \Pi_S \Phi$. PPS also reveals a structural limitation of output-based interpretability: latent components in $\ker(\Pi)$ 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.
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 $(H, \Pi, O)$, where $H$ is a latent representation space, $\Pi \succeq 0$ is an observation operator, and $O(v)=\langle v,\Pi v\rangle$ defines an induced scalar observable. Observability is characterized by the quotient geometry $H/\ker(\Pi)$, 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 $\Phi \Pi_T \approx \Pi_S \Phi$. PPS also reveals a structural limitation of output-based interpretability: latent components in $\ker(\Pi)$ 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.
29 pages, 7 figures. Published in Entropy
Machine Learning (cs.LG)
arXiv:2607.05175 [cs.LG]
(or arXiv:2607.05175v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2607.05175 Focus to learn more arXiv-issued DOI via DataCite
https://doi.org/10.3390/e28070768 Focus to learn more DOI(s) linking to related resources
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Full text under the publisher's open licence · 326 words
