arXiv：cs.LG· Kazuo Ishii, Bishnu Prasad Gautam, Jieling Wu, Saher Javaid·· 2 小时前
Platonic Projection Structures：表示学习中的算子诱导可观测性
Platonic Projection Structures: Operator-Induced Observability in Representation Learning
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研究提出 Platonic Projection Structures（PPS）框架，用作用于潜在表示空间的自伴半正定算子刻画部分观测下的表示可观测性，将系统表示为三元组 (H, Π, O)，并以商几何 H/ker(Π) 描述观测下不可区分的等价类。
正文
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.
| Comments: | 29 pages, 7 figures. Published in Entropy |
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2607.05175 [cs.LG] |
| (or arXiv:2607.05175v2 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2607.05175 arXiv-issued DOI via DataCite |
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| Journal reference: | Entropy 2026, 28(7), 768 |
| Related DOI: | https://doi.org/10.3390/e28070768
DOI(s) linking to related resources |
Submission history
From: Kazuo Ishii [view email]
[v1]
Mon, 6 Jul 2026 14:57:26 UTC (1,418 KB)
[v2]
Wed, 14 Oct 2026 03:01:50 UTC (1,418 KB)
来源：arXiv：cs.LG · arxiv.org