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The MapDiff denoising diffusion framework iteratively alternates between two processes: diffusion and denoising. The diffusion process progressively adds random discrete noise to the native sequence {{\bf{X}}}_{0}^{\rm{aa}} according to the cumulative transition matrix {\overline{\bf{Q}}}_{t} at the diffusion step t so that the real data distribution can gradually transition to a uniform or marginal prior distribution. The denoising process randomly samples an initial noisy AA sequence {{\bf{X}}}_{T}^{\rm{aa}} from the prior distribution and iteratively uses the denoising network ϕ θ in b to denoise it, learning to predict the native sequence {{\bf{X}}}_{0}^{\rm{aa}} from {{\bf{X}}}_{t}^{\rm{aa}} at each denoising step t . The prediction {\hat{{\bf{X}}}}_{0}^{\rm{aa}} facilitates the computation of the posterior distribution q({{\bf{X}}}_{t-1}^{\rm{aa}}| {{\bf{X}}}_{t}^{\rm{aa}},{\hat{{\bf{X}}}}_{0}^{\rm{aa}}) for predicting a less-noisy sequence {{\bf{X}}}_{t-1}^{\rm{aa}}.

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