PaperSummary01 : Deep unsupervised learning using nonequilibrium thermodynamics

The paper introduces diffusion probabilistic models which allow extreme flexibility in model structure, exact sampling, easy multiplication with other distributions, cheap evaluation of model log likelihood and the probability of individual states.

The proposed method uses a Markov chain to gradually convert one distribution into another ( the idea is inspired by non equilibrium in statistical physics and sequential Monte Carlo).

The proposed algorithm describes the following steps:

1. The forward trajectory refers to the process of gradually transforming the complex data distribution into a simple, analytically tractable distribution (such as a Gaussian) through a diffusion process. This transformation is achieved via repeated applications of a Markov diffusion kernel over multiple time steps, enabling the model to simplify the data’s structure progressively .
2. The reverse trajectory is the process of learning a generative Markov chain that starts from the simple, analytically tractable distribution (e.g., a Gaussian) and reconstructs the original complex data distribution. This is achieved by training the model to reverse the forward diffusion process step-by-step, using learned mean and covariance functions to gradually restore the data’s structure .
3. The model probability is the likelihood that the generative model assigns to the observed data. It is calculated by integrating over all possible latent trajectories in the reverse diffusion process, balancing the probabilities of the forward and reverse trajectories. This can be efficiently estimated using techniques like annealed importance sampling and averaging over samples from the forward trajectory to account for the quasi-static nature of the process .
4. Training involves maximizing a lower bound on the model’s log-likelihood, ensuring the reverse diffusion process accurately reconstructs the original data distribution. This is achieved by minimizing the Kullback-Leibler (KL) divergence between the forward trajectory’s conditional distributions and the reverse trajectory, while also estimating the reverse diffusion kernel parameters such as mean and covariance for Gaussian diffusion or bit-flip probabilities for binomial diffusion .
5. To combine the learned model distribution with an additional function to compute posteriors or perform tasks like denoising and inpainting is achieved by modifying the reverse diffusion trajectory, incorporating at each step, while preserving the tractable forms of the diffusion kernels .
6. Entropy of the reverse process is derived ( lower and upper bounds) of each step.

References:

Deep Unsupervised Learning using Nonequilibrium Thermodynamics