Source
Johnson, El Hanchi & Maddison (2023), LIT-258, Thm 3.2 and App. E.2; Tosh, Krishnamurthy & Hsu (2021), LIT-251, §2.4; HaoChen et al. (2021), LIT-249, App. F (Assumption F.1, Thm F.2, eq. 70). The integrability condition and its identification with finite χ² are the record’s own.
What was actually shown
The derivation. For the positive-pair distribution with marginal p, (Tf)(a) = E[f(A′) | A = a] = ∫ p(b|a) f(b) db = ∫ K⁺(a,b) f(b) p(b) db, with K⁺ = p⁺/(p⊗p). So K⁺ is exactly the kernel of the conditional-expectation operator on L²(p). T is Hilbert–Schmidt, hence compact with a discrete spectrum, iff ∫∫ K⁺² dp dp < ∞, and that double integral equals 1 + χ²(P⁺‖P⊗P). The positive-pair symmetry makes T self-adjoint.
What the sources show, which this reading unifies: Johnson et al.’s eigenfunctions (Thm 3.2) are T’s eigenfunctions on a finite space, and they write the first step of the derivation themselves: λ_i f_i(a₁) = Σ K⁺(a₁,a₂) f_i(a₂) p(a₂) = E_(a₂|a₁)[f_i(a₂)] (App. E.2), with the integral form given as motivation and no integrability condition. Tosh et al. read the density ratio as a change of measure and the infer-then-predict function µ(x) = E[E[Y|Z]|X=x], which is within εX + 2√(εXεZ) + εZ of the Bayes predictor under redundancy (Lemma 1), as an integral against E[Y|Z] (§2.4); calling it an operator is the record’s. Their landmark bound is controlled by var(E[Y|Z]g*(X,Z̃)) ≤ E[g*²] = 1 + χ², the same Hilbert–Schmidt quantity (the identity with χ² is the record’s algebra), but their Theorems 10–11 need a uniform bound g_max on the ratio, like HaoChen et al.’s F.1. HaoChen et al. extend to infinite spaces under Assumption F.1, which requires finite marginals and a uniform bound w(u,v)/w(u) ≤ B·w(v), that is, K⁺ ≤ B everywhere. That is strictly stronger than finite χ². They derive the Hilbert–Schmidt property from it (eq. 70) and call it sufficient, not necessary. They prove only that an eigenbasis exists (Thm F.2); the extension of Theorems 3.8, 4.2 and 4.3 is asserted with the proof omitted. That the Hilbert–Schmidt integral is 1 + χ²(P⁺‖P⊗P), and that finite χ² is the condition the argument needs, is the record’s inference, not HaoChen et al.’s.
The L²(p) here is the space the commutative GNS construction produces from the state f ↦ E_p f (Jorgensen & Tian, LIT-259, Remark 1.34 and Ex. 4.28, say the kernel-to-Hilbert-space construction is GNS and that commutative states are measures). Ex. 4.86 and Lemma 3.28 state outright that the commutative GNS space is L²(μ), but only for compact X (or a compact spectrum); for the R^d spaces common here the passages read say nothing, so the identification there is still the record’s inference.
What this does not say
- That InfoNCE-trained features span T’s top eigenfunctions. InfoNCE fits log K⁺, not K⁺, so its embeddings’ inner products are not T’s kernel.
- That HaoChen et al.’s guarantees hold under finite χ² alone. Their infinite-space case assumes the uniform bound of Assumption F.1, and its extension of the main theorems is not written out.
- Anything when χ²(P⁺‖P⊗P) is infinite, which happens when views are too informative about each other (for example, near-deterministic augmentations on continuous inputs).
- That the GNS identification adds a result. It names the space; the theorems are the spectral theorem and the sources’ own.
Note of 2026-10-19: the derivation filed as the record’s own
The derivation in “What was actually shown”, that K⁺ is T’s kernel, that T is Hilbert–Schmidt with ‖T‖²_HS = 1 + χ², and that it is self-adjoint, is now CLAIM-370. It is stated there with a proof, including the converse (Hilbert–Schmidt only if χ² is finite), and it is Active. Mathematics the record proves does not wait on a source. What this THEORY still waits on is what the sources show, and whether features reach the eigenfunctions.