Cauchy Schedule

The Cauchy schedule draws its shape from the heavy-tailed Cauchy probability distribution. Noise reaches its maximum rate of change near the center ($x_0$), with a sharp transition at the peak. Unlike smoother schedules such as cosine, the Cauchy distribution introduces more localized variations. Its heavy tails assign greater weight near the peak while maintaining meaningful noise variation far from the center. This makes the Cauchy schedule a preferred choice when noise needs to be concentrated within a specific range of timesteps, allowing controlled noise generation over the diffusion process.