CLAIM-370: For a symmetric positive-pair distribution with marginal p, the conditional-expectation operator on L²(p) has the density ratio K⁺ as its kernel, and it is Hilbert–Schmidt exactly when the χ²-divergence of the pair from the product of its marginals is finite, in which case it is compact and self-adjoint with an orthonormal eigenbasis

The claim

Let (A, A′) take values in a standard Borel space X, with joint distribution P⁺ symmetric under exchange of the two views and marginal p. Define T on L²(p) by (Tf)(a) = E[f(A′) | A = a].

1. Kernel. If P⁺ ≪ p⊗p, with density K⁺ = dP⁺/d(p⊗p), then (Tf)(a) = ∫ K⁺(a, b) f(b) p(db).
2. The exact condition. T is Hilbert–Schmidt if and only if P⁺ ≪ p⊗p and χ²(P⁺‖p⊗p) < ∞, and then ‖T‖²_HS = ∫∫ K⁺² d(p⊗p) = 1 + χ²(P⁺‖p⊗p).
3. Spectrum. T is self-adjoint and a contraction. When it is Hilbert–Schmidt it is compact, so L²(p) has an orthonormal basis of eigenfunctions of T together with its kernel. The eigenvalues are real, lie in [−1, 1] and accumulate only at 0, and the constants are an eigenfunction with eigenvalue 1.

Proof

(1) For bounded g, E[g(A)f(A′)] = ∫∫ g(a)f(b)K⁺(a,b) p(da)p(db), and this identifies the conditional expectation as the stated integral.

(2) Hilbert–Schmidt operators on L²(p) are exactly the integral operators with a kernel in L²(p⊗p). If K⁺ ∈ L²(p⊗p), then (1) shows that T is one. Conversely, let T be Hilbert–Schmidt with kernel k. For measurable E and F, P⁺(E × F) = ⟨1_E, T1_F⟩ = ∫∫ 1_E(a)1_F(b)k(a,b) p(da)p(db). Measures that agree on rectangles agree, so P⁺ = k·(p⊗p) and k = K⁺. In both directions, ∫∫ K⁺² d(p⊗p) = ∫ (dP⁺/d(p⊗p))² d(p⊗p) = 1 + χ²(P⁺‖p⊗p), by the definition of χ².

(3) ⟨g, Tf⟩ = E[g(A)f(A′)], which is symmetric in f and g because P⁺ is. By Jensen, (Tf)² ≤ T(f²), so ‖Tf‖² ≤ E[f(A′)²] = ‖f‖²; T1 = 1. A compact self-adjoint operator has the stated spectral decomposition (the spectral theorem).

What it does not say

• That trained features span T’s top eigenfunctions. That is about optimization and capacity. THEORY-005’s promote_when keeps it.
• That InfoNCE features are eigenfunctions. InfoNCE fits log K⁺, not K⁺ (THEORY-001).
• Anything when χ² is infinite, for instance when two views of a continuous input are nearly deterministic functions of each other. Then T need not be compact. In the limit A′ = A, T is the identity.
• That a source states it. Johnson et al. (LIT-258, App. E.2) write (1) on a finite space, where (2) is automatic. HaoChen et al. (LIT-249, Assumption F.1, eq. 70) get the Hilbert–Schmidt property from K⁺ ≤ B, which is sufficient and strictly stronger.