Source
Peyré, Cuturi and Solomon (2016), LIT-1797, §§2.3–3.1, pp. 2667–2669 of the ICML 2016 proceedings; read in NOTE-1343.
What was actually shown
The closed form (Prop. 3, p. 2668). For a loss L(a, b) = f₁(a) + f₂(b) − h₁(a)h₂(b) with f₁′/h₁′ invertible, the C step minimizing Σ λ_s ⟨L(C, C_s) ⊗ T_s, T_s⟩ is C = (f₁′/h₁′)⁻¹(Σ λ_s (realigned h₂(C_s)) / ppᵀ). This is a global minimizer when L is convex in its first argument (p. 2668). The realigned matrix is T_s h₂(C_s) T_sᵀ for T_s ∈ ℝ^{N×N_s}. The paper prints T_sᵀ … T_s, which does not fit its own convention.
Square loss and KL (Props. 4–5, pp. 2668–2669). For square loss the update is the weighted arithmetic average of the realigned C_s, divided entry-wise by ppᵀ. A positive combination of PSD matrices with a positive diagonal rescaling is PSD, so PSD inputs give PSD iterates. For KL the update is the exponential of the weighted average of the realigned log C_s, and conditional positive-definiteness is preserved, so infinitely divisible inputs give infinitely divisible iterates. Checked: 200 random trials with PSD inputs of sizes 3, 7 and 4 gave a smallest eigenvalue of at least 0.009.
What is not guaranteed. The coupling step converges for step sizes below a τ_max, but the step τ = 1/ε that makes it a Sinkhorn iteration “is not covered by the theory” (Remark 3, p. 2667). The barycenter problem “is convex with respect to C but not with respect to (Ts )s” (p. 2668). Its size N and histogram p are “fixed by the user” (p. 2668).
What could have failed. The closed-form average could have left the PSD or infinitely divisible cone, which would make the barycenter of kernels not a kernel. The propositions rule that out.
What this does not say
- That a minority’s structure is erased. With couplings fixed it is shrunk by its weight λ_s. In a toy run (three two-cluster inputs, one with a split cluster, λ = ¼, N = 8), the optimized barycenters landed on 9 distinct objective values over 20 seeds. The split survived at sizes between 2.20 and 5.24, against 0.92 for the fixed-alignment average. What the optimization does to a minority distinction depends on the initialization and on N, p, λ and ε.
- That the barycenter is unique, or the same across initializations.
- That the barycenter is the structure common to the inputs. It is a Fréchet mean under GW_ε for weights and a size chosen in advance.