Platonic Projection Structures: Operator-Induced Observability in Representation Learning

Modern machine learning systems are often interpreted through their outputs, such as predictions, feature activations, or attribution maps. However, these observable quantities rarely provide a complete view of the underlying latent representations. This paper introduces Platonic Projection Structures (PPS), an operator-theoretic framework that formalizes this limitation. Instead of assuming that outputs are direct reflections of latent states, PPS models observation as a geometric process where a latent space is projected through a self-adjoint positive semidefinite operator. This approach reveals that observability is not an intrinsic property of the data, but a structural consequence of the observation operator itself.

At the heart of PPS is the concept of a quotient geometry. The observation operator induces an equivalence relation where latent states that differ only by a component in the operator's kernel are indistinguishable. Consequently, the effective observable space is the quotient space of the latent representation space by this kernel. This implies that any information contained within the kernel is structurally inaccessible to downstream tasks, including post hoc interpretability methods. The framework demonstrates that this is not an algorithmic failure, but a geometric constraint inherent to the projection process.

PPS provides a unified language for analyzing observability across different domains. The authors show that both quantum measurement and deep learning inference share this operator-theoretic structure. While quantum systems utilize orthogonal projections, deep learning models with linear readouts induce non-idempotent positive semidefinite operators. By framing both as projection-mediated observation, the authors offer a consistent way to analyze representation transfer and knowledge distillation. Specifically, they interpret distillation as the approximate preservation of observable geometry between teacher and student models, proposing an operator-consistency objective that aligns these geometries.