Platonic Projection Structures: Operator-Induced Observability in Representation Learning

Area of Science:

• Machine Learning
• Representation Learning
• Operator Theory
• Quantum Information Theory

Background:

• Representation learning aims to uncover latent structures from data.
• Partial observation poses challenges for understanding and interpreting these latent representations.
• Existing frameworks often treat observations as direct reflections of latent states.

Purpose of the Study:

• To introduce Platonic Projection Structures (PPS), an operator-theoretic framework for characterizing observability in representation learning.
• To unify the understanding of representation accessibility, interpretability, and transfer under partial observation.
• To reveal fundamental limitations of output-based interpretability inherent in the observation geometry.

Main Methods:

• Developed an operator-theoretic framework (PPS) modeling observation as a geometry induced by an operator on a latent Hilbert space.
• Characterized observability via quotient geometry (H/ker(Π)), representing indistinguishable latent states.
• Formulated quantum measurement and linear observation models within the PPS structure.
• Analyzed representation transfer and knowledge distillation using the intertwining condition.
• Provided empirical validations for kernel-invariant observability and projection-induced attribution gaps.

Main Results:

• Established a unified operator-theoretic structure for quantum measurement and linear representation inference.
• Demonstrated that observable behavior is governed by the geometry induced by the observation operator, not latent representations directly.
• Revealed that latent components within the kernel of the observation operator (ker(Π)) are fundamentally inaccessible.
• Showcased that interpretability methods inherit constraints from the observation geometry, leading to intrinsic attribution gaps.
• Empirically validated kernel-invariant observability and rank-controlled observable geometry.

Conclusions:

• Platonic Projection Structures (PPS) provide a mathematically explicit characterization of observability through operator-induced quotient geometry.
• The framework offers a unified perspective on representation accessibility, interpretability, and representation transfer.
• Output-based interpretability methods have inherent structural limitations imposed by the observation process itself.