Answers.Body: <p>Since in this ideal case well-trained probability flow ODEs by score functions are essentially same as normalizing flow models with well-defined bijective transformations, the quality of generated image should in general as good as normalizing flow models' result as long as the cumulative numerical errors during deterministic transformations computation are bounded and negligible (the referenced answer talked about why injecting stochastic Langevin diffusion noise would actually correct these cumulative errors on average) <em>and</em> your sampling is from the high probability regions of the disentangled Gaussian latent space. Most of the image's qualities are already encoded in the reverse bijective transformations, similar to DDPM's reverse diffusion random sampling process originated from the same final-layer's Gaussian latent space to reconstruct the input data after training phase of the amortized variational noises is done successfully.</p>
  
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Body: <p>I have read the question and the given answer <a href="https://ai.stackexchange.com/questions/46188/does-probability-flow-ode-trajectory-in-the-context-of-diffusion-models-repres">Does probability flow ODE trajectory (in the context of diffusion models) represents a bijective mapping between any distribution to a gaussian?</a> <br />
  Now I have a follow up question: Let's assume we are talking about the ideal case, e.g. we have access to <span class="math-container">$T=\infty$</span>
  Imagine we have learnt the score function associated with distribution P. <br />
  Now, imagine we take an (clean) image from the distribution <span class="math-container">$P_0$</span> and then we follow the probability flow ode trajectory (we don't inject noise). At <span class="math-container">$T=\infty$</span> , the image will be encoded to a gaussian distribution of the same dimension (let's say d-dimensional). <br />
  My question is: Can we freely change any of those d elements (independent of anything), and then follow the reverse trajectory and get an image from distribution <span class="math-container">$P_0$</span> ? In the other word, if we start from any point in the gaussian and follow the reverse ode (not SDE/ no noise injection) do we end up in a good looking image with a high probability?</p>
  
Tags: <generative-model><diffusion-models>
Title: Does ideal probability flow-ode models generate good quality data starting from any (high probable) points in the Gaussian distribution?
UserDisplayName: saleh
ThreadText: # Does ideal probability flow-ode models generate good quality data starting from any (high probable) points in the Gaussian distribution?
  
  (Asked by: username_0 on 2024-07-17)
  
  I have read the question and the given answer [Does probability flow ODE trajectory (in the context of diffusion models) represents a bijective mapping between any distribution to a gaussian?](https://ai.stackexchange.com/questions/46188/does-probability-flow-ode-trajectory-in-the-context-of-diffusion-models-repres)   
  Now I have a follow up question: Let's assume we are talking about the ideal case, e.g. we have access to $T=\infty$
  Imagine we have learnt the score function associated with distribution P.   
  Now, imagine we take an (clean) image from the distribution $P\_0$ and then we follow the probability flow ode trajectory (we don't inject noise). At $T=\infty$ , the image will be encoded to a gaussian distribution of the same dimension (let's say d-dimensional).   
  My question is: Can we freely change any of those d elements (independent of anything), and then follow the reverse trajectory and get an image from distribution $P\_0$ ? In the other word, if we start from any point in the gaussian and follow the reverse ode (not SDE/ no noise injection) do we end up in a good looking image with a high probability?
  
  **Tags:** generative-model, diffusion-models
  
  
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  ## 1 answers
  
  ### Answer by username_1 on 2024-07-20. Score: 0 | Accepted answer
  
  Since in this ideal case well-trained probability flow ODEs by score functions are essentially same as normalizing flow models with well-defined bijective transformations, the quality of generated image should in general as good as normalizing flow models' result as long as the cumulative numerical errors during deterministic transformations computation are bounded and negligible (the referenced answer talked about why injecting stochastic Langevin diffusion noise would actually correct these cumulative errors on average) *and* your sampling is from the high probability regions of the disentangled Gaussian latent space. Most of the image's qualities are already encoded in the reverse bijective transformations, similar to DDPM's reverse diffusion random sampling process originated from the same final-layer's Gaussian latent space to reconstruct the input data after training phase of the amortized variational noises is done successfully.
  
  
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