Physics-Aware Learnability: From Set-Theoretic Independence to Operational Constraints
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917300614987776 |
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| author | Bang, Jeongho Cho, Kyoungho |
| author_facet | Bang, Jeongho Cho, Kyoungho |
| contents | Beyond binary classification, learnability can become a logically fragile notion: in EMX, even the class of all finite subsets of $[0,1]$ is learnable in some models of ZFC and not in others. We argue the paradox is operational. The standard definitions quantify over arbitrary set-theoretic learners that implicitly assume non-operational resources (infinite precision, unphysical data access, and non-representable outputs). We introduce physics-aware learnability (PL), which defines the learnability relative to an explicit access model -- a family of admissible physical protocols. Finite-precision coarse-graining reduces continuum EMX to a countable problem, via an exact pushforward/pullback reduction that preserves the EMX objective, making the independence example provably learnable with explicit $(ε,δ)$ sample complexity. For quantum data, admissible learners are exactly POVMs on $d$ copies, turning sample size into copy complexity and yielding Helstrom(-type) lower bounds. For finite no-signaling and quantum models, PL feasibility becomes linear or semidefinite and is therefore decidable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00417 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Physics-Aware Learnability: From Set-Theoretic Independence to Operational Constraints Bang, Jeongho Cho, Kyoungho Machine Learning Quantum Physics Beyond binary classification, learnability can become a logically fragile notion: in EMX, even the class of all finite subsets of $[0,1]$ is learnable in some models of ZFC and not in others. We argue the paradox is operational. The standard definitions quantify over arbitrary set-theoretic learners that implicitly assume non-operational resources (infinite precision, unphysical data access, and non-representable outputs). We introduce physics-aware learnability (PL), which defines the learnability relative to an explicit access model -- a family of admissible physical protocols. Finite-precision coarse-graining reduces continuum EMX to a countable problem, via an exact pushforward/pullback reduction that preserves the EMX objective, making the independence example provably learnable with explicit $(ε,δ)$ sample complexity. For quantum data, admissible learners are exactly POVMs on $d$ copies, turning sample size into copy complexity and yielding Helstrom(-type) lower bounds. For finite no-signaling and quantum models, PL feasibility becomes linear or semidefinite and is therefore decidable. |
| title | Physics-Aware Learnability: From Set-Theoretic Independence to Operational Constraints |
| topic | Machine Learning Quantum Physics |
| url | https://arxiv.org/abs/2603.00417 |