[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83911-en":3,"doc-seo-83911-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},83911,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Platonic Projection Structures Operator-Induced Observability in Representation Learning","Platonic Projection Structures (PPS) characterizes observability in representation learning using an operator-theoretic framework for how latent representations become accessible under partial observation. PPS models observation via a self-adjoint positive semidefinite operator that induces geometry on a latent Hilbert space, forming a system triple (H, Π, O) with O(v)=⟨v,Πv⟩. Observability is governed by the quotient geometry H/ker(Π), where kernel components are indistinguishable and structurally inaccessible. PPS unifies quantum measurement and linear observation inference, clarifies attribution limits, and provides empirical validations on kernel-invariant behavior and rank-controlled observable geometry.","arXiv :2607 .05 175v 1 [ cs .LG] 6 Jul 2026  \nPlatonic Projection Structures: Operator-Induced Observability in  \nRepresentation Learning  \nKazuo Ishii 1 ,∗ , Bishnu Prasad Gautam 1 , Jieling Wu 1 , and Javaid Saher2  \n1 Department of Applied Information Engineering, Faculty of Engineering, Suwa University of Science,  \nChino 391-0292, Nagano, Japan  \n2 Department of Information Engineering, Kanazawa Gakuin University,  \nKanazawa 920-1392, Ishikawa, Japan  \n∗ Corresponding author: [kishii@rs.sus.ac.jp](kishii@rs.sus.ac.jp)  \nAbstract  \nWe 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 as a geometry induced by a self-adjoint positive semidefinite operator acting on a latent representation space. A system is represented as a triple (H, Π, O), where H denotes a latent representation space, Π ⪰ 0 is an observation operator, and O (v) = ⟨v,Πv⟩ defines an induced scalar observable. The framework characterizes observability through the quotient geometry H/ker(Π), which represents equivalence classes of latent states that are indistinguishable under observation. From this perspective, observable behavior is governed not by latent representations themselves, but by the geometry induced through the observation operator. We show that both quantum measurement and representation inference under linear observation models can be formulated within this common operator-theoretic structure while differing in the algebraic properties of their observation operators. Within this perspective, quantum measurement serves primarily as a mathematically canonical example of projectionmediated observability. The correspondence developed in PPS is therefore structural rather than physical. Within the same framework, representation transfer and knowledge distillation can be interpreted as approximate preservation of observable geometry through the intertwining condition ΦΠT ≈ ΠS Φ . PPS further reveals a structural limitation of output-based interpretability:  \nlatent components contained in ker(Π) are fundamentally inaccessible from observables generated through the induced observation process. Accordingly, attribution and explanation methods inherit intrinsic constraints imposed by the observation geometry itself. We provide controlled empirical validations demonstrating kernel-invariant observability, projection-induced attribution gaps, and rank-controlled observable geometry in latent representation spaces. Overall, PPS provides a mathematically explicit characterization of observability through operator-induced quotient geometry, offering a unified perspective on representation accessibility, interpretability, and projection-mediated inference.  \nKeywords: observability; representation learning; operator-theoretic framework; positive semidefinite operators; knowledge distillation; interpretability; quotient geometry; quantum measurement  \n1 Introduction  \nModern machine learning systems are increasingly interpreted through observable quantities such as output distributions, attribution maps, representations, and feature activations [1–3] .  \nAt the same time, recent developments in information-theoretic analysis have highlighted that observable outputs often provide only partial access to the underlying latent structure of a system [8– 11] . Despite substantial progress in explainable artificial intelligence (XAI), representation analysis, and information bottleneck approaches [1–3 , 8–11], the notion of observability itself remains largely implicit rather than explicitly formalized in modern learning systems. In many existing formulations, observable outputs are implicitly treated as sufficiently informative reflections of latent representations. However, practical learning systems frequ","cbCaidgUCk7Xh9Tl","https://ap.wps.com/l/cbCaidgUCk7Xh9Tl","pdf",2171231,1,29,"English","en",105,"# Abstract\n# Introduction\n## PPS system formulation (H, Π, O)\n## Induced equivalence relation and quotient geometry\n## Inaccessible latent components and spectral characterization","[{\"question\":\"What does PPS use to model observation in representation learning?\",\"answer\":\"PPS models observation as a geometry induced by a self-adjoint positive semidefinite operator Π acting on a latent representation space, rather than treating observed outputs as direct reflections of latent states.\"},{\"question\":\"How is observability characterized in PPS?\",\"answer\":\"Observability is determined by the quotient geometry H/ker(Π), where vectors differing by elements in ker(Π) are indistinguishable under the induced observable quantity O(v)=⟨v,Πv⟩.\"},{\"question\":\"Why do attribution and explanation methods have intrinsic limitations in PPS?\",\"answer\":\"PPS shows that latent components contained in ker(Π) are fundamentally inaccessible from observables generated by the induced observation process, so attribution methods inherit constraints imposed by the observation 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does PPS use to model observation in representation learning?","Question",{"text":75,"@type":76},"PPS models observation as a geometry induced by a self-adjoint positive semidefinite operator Π acting on a latent representation space, rather than treating observed outputs as direct reflections of latent states.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is observability characterized in PPS?",{"text":80,"@type":76},"Observability is determined by the quotient geometry H/ker(Π), where vectors differing by elements in ker(Π) are indistinguishable under the induced observable quantity O(v)=⟨v,Πv⟩.",{"name":82,"@type":73,"acceptedAnswer":83},"Why do attribution and explanation methods have intrinsic limitations in PPS?",{"text":84,"@type":76},"PPS shows that latent components contained in ker(Π) are fundamentally inaccessible from observables generated by the induced observation process, so attribution methods inherit constraints imposed by the observation 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