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Langevin Dynamics

Starting with a sample from some initial distribution \(\tilde{x}_{0}\sim \pi(x)\), we can recursively apply the Langevin method to compute:

\[ \tilde{x}_t=\tilde{x}_{t-1}+\frac{\epsilon}{2}\nabla_{x}\log{p(\tilde{x}_{t-1})}+\sqrt{\epsilon}\space z_{t} \]

where

• \(\epsilon>0\) is the step-size
• \(z_{t}\sim \mathcal{N}(0,I)\) is random noise Given a small enough step size \(\epsilon\) and large number of steps \(T\), this method will produce samples from \(p_{data}(x)\) using only the trajectory described by the score function \(\nabla_{x}\log{p(x)}\).