Organizations: Department of Applied Information Engineering, Faculty of Engineering, Suwa University of Science, Chino 391-0292, Nagano, Japan · Department of Information Engineering, Kanazawa Gakuin University, Kanazawa 920-1392, Ishikawa, Japan
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⪰0 is an observation operator, and O(v)=⟨v,Πv⟩ 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 ΦΠT≈ΠSΦ. 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.
Identifying latent state representations and dynamics is essential when direct modeling in observation space is infeasible, particularly under partial and high-dimensional observations. In such settings, representation learning and physics-aware modeling are inherently coupled. We study this problem for latent port-Hamiltonian systems, a structured class encompassing both conservative and dissipative dynamics. We propose a two-stage identify-then-project framework. First, a contrastive teacher learns continuous-time latent dynamics from partial observations. Then, a student projects the identified teacher representation and dynamics onto a port-Hamiltonian submanifold via a learned affine chart, yielding a physically consistent realization. As a conceptual counterfactual, we also consider a single-stage variant that jointly learns latent identification and port-Hamiltonian structure, but find it to be less reliable, motivating the proposed two-stage teacher-student framework. We show theoretically that affine projection is the natural bridge between the affine gauge of contrastive latent identification and the port-Hamiltonian systems. Empirically, we demonstrate that the proposed two-stage approach preserves the teacher's dynamics while enforcing physical structure, and performs more reliably than the single-stage alternative, particularly in dissipative regimes and high-dimensional visual settings.
Learned representations in intelligent sensing systems are often evaluated by reconstruction fidelity or downstream prediction accuracy, but these criteria do not specify which latent distinctions are justified by the sensing process. In sensor-conditioned environments, nuisance factors can change measurements without changing the scene, while distinct scenes may be indistinguishable under limited sensing capability. This paper formulates sensor-conditioned representation correctness as preserving sensing-supported scene distinctions while suppressing nuisance-induced and sensor-unsupported variation. We introduce the scene-relevant observation quotient, a representation target induced by sensing-supported distinguishability after nuisance canonicalization, and develop Observation-Quotient Tucker-Structured Autoencoding (OQ-TSAE), a scene-nuisance factorized framework with diagnostics for false distinction, false merge, nuisance sensitivity, and latent ordering consistency. Experiments on a controlled benchmark show that quotient-consistent supervision improves representation-correctness diagnostics over reconstruction-oriented, metric-learning, and contrastive-learning baselines. Sensitivity, perturbation, and ablation studies show the importance of quotient-aligned supervision, reliable quotient relations, and quotient geometry. Complementary real-radar experiments show that a reconstruction-only OQ-TSAE variant retains competitive downstream utility, robustness under observation degradation, and low seed-to-seed variability. These results suggest that sensor-conditioned representations should be evaluated not only by predictive utility, but also by whether their latent geometry preserves sensing-justified scene distinctions.
Inference and control in engineered physical systems pay a heavy physics cost at deployment: state estimators, inverse-problem solvers, model-predictive controllers, schedulers, and observers are often not closed-form and must re-solve a numerical optimization per instance, with the operator re-supplied each time. Physics-informed learning moves this cost to training, but uses a single encoder pathway whose latent geometry de-learns under fine-tuning and admits no quantitative transfer guarantee. We propose an asymmetric two-pathway architecture that resolves both issues. A teacher encoder consumes privileged dense states from a high-fidelity simulator and represents the system through operator-polynomial features stable under spectral perturbation; a student encoder learns the same latent geometry from sparse field data and operator descriptors. At deployment the teacher is discarded, and the frozen student runs in a single forward pass with a transfer certificate. The design connects to privileged-information learning, knowledge distillation, and cross-modal distillation, but targets cross-instance transfer rather than fixed-instance prediction: topology and operator may change, while the latent task does not. We establish sufficient and near-necessary transfer conditions via Wasserstein proximity between latent laws, yielding a zero-shot error bound, and develop a finite-sample certification protocol with active expansion when coverage is incomplete. The framework applies wherever a system admits an operator with reportable spectrum. On power-system estimation, it achieves zero-shot transfer to 100 unseen topologies, a 95% certificate pass rate, accuracy competitive with topology-aware Newton--Raphson, and sub-millisecond inference. These results suggest asymmetric pathways plus operator-anchored latent geometry provide a foundation for certified zero-shot inference and control.