Multitask Learning with Stochastic Interpolants

The paper’s diffusion claim (Proposition 2.6) hinges on Lemma 2.4, which states η0(α,β,x) = −α sα,β(x) and hence sα,β(x) = −α−1η0(α,β,x) when x0 ~ N(0,Id). This omits a transpose: the correct Gaussian integration-by-parts identity is sα,β(x) = −α−TΣ0−1η0(α,β,x) (with Σ0 the covariance of x0; for Σ0 = I this reduces to s = −α−Tη0). Using the paper’s incorrect s = −α−1η0 leads to the SDE drift (α̇t − εtα−1t)η0 that does not preserve the intended marginals for non-symmetric αt. A concrete 2×2 counterexample shows the mismatch. By contrast, replacing α−1 by α−TΣ0−1 in the drift yields an SDE whose time marginals match the ODE, as the model proves via an Itô–Fokker–Planck cancellation using the corrected score identity. The paper’s ODE proposition (Proposition 2.5) is fine; the error is confined to the score identity and the diffusion Proposition 2.6. See Lemma 2.4 and its proof (eq. (26)) and Proposition 2.6 (eq. (8)) in the paper’s text and appendix for the exact statements the model corrects .