AI summaryⓘ
The authors introduce Platonic Projection Structures (PPS) as a new way to understand how we can observe or access hidden features in learned representations when only partial data is available. Instead of assuming observations directly show the hidden details, PPS uses mathematical operators to describe what can be detected and what remains hidden. They show that this approach clarifies the limits of methods like interpretability, knowledge transfer, and how observations relate to quantum measurements in a purely structural way. Their experiments demonstrate how some parts of the hidden information are fundamentally unobservable, which affects how well we can explain or transfer learned knowledge. Overall, the authors provide a framework to better understand the geometry and limitations of observing latent representations.
observabilityrepresentation learningoperator theorylatent spaceself-adjoint operatorquotient geometrykernel (linear algebra)knowledge distillationinterpretabilityquantum measurement
Authors
Kazuo Ishii, Bishnu Prasad Gautam, Jieling Wu, Javaid Saher
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.