Generative Fractional Interpolants

Mandelbrot–Van Ness fractional Brownian motion (Type I fBM) with Hurst index $H\in (0,1)$ interpolates between Brownian motion, which drives continuous-time diffusion models, and random straight-line paths used in flow matching as $H\to 1$. The Hurst index controls the correlation of increments and thereby the temporal memory of the noise process. Away from $H=1/2$, this memory makes fBM non-Markovian, motivating Markovian approximations for tractable computation. However, the Markovian approximation of fBM (MA-fBM) used in prior work exhibits increasing error as $H\to 1$ and fails to preserve the random straight-line limit. We therefore improve MA-fBM by augmenting the approximation process with the terminal value of fBM and derive the optimal approximation coefficients in closed form. The resulting $\epsilon$-MA-fBM preserves this limiting behavior while reducing the approximation error. Building on this approximation, we introduce *Generative Fractional Interpolants (GFI)*, a generative model interpolating subdiffusive dynamics ($H<1/2$), standard diffusion ($H=1/2$), superdiffusive dynamics ($H>1/2$), and random straight line paths ($H\to 1$) used in flow matching. To train our generative model, we derive a KL-divergence of the reverse process and its parameterization and train over the full range of Hurst indicies, allowing the best operating regime to be selected at sampling time. Notably, unlike flow matching, GFI can achieve straight generative trajectories from noise to data, at the cost of trajectory crossings. Empirically, this design choice yields in our experiments lower FD-DINOv2 scores than diffusion models and flow matching on CIFAR-10, the imbalanced CIFAR-10-LT dataset, and CelebA-HQ-256 in latent space.