Towards demystifying the creativity of diffusion models

Score smoothing facilitates manifold recovery

In the real world, complex data like high-resolution images live in high-dimensional pixel spaces rather than a simple 1-D world. The vast majority of that space, however, is just random noise that is meaningless to the human eye. Only a small fraction of the data points in that space correspond to recognizable images, and they live in what’s called the data manifold (like a sheet tucked inside a larger space). The shape and location of the data manifold are not known by the model in advance. Thus, image generation can be considered as a task of manifold recovery, where the model needs to infer what the hidden data manifold looks like based on the finite number of training data sampled from it, and then come up with new points on the manifold which will correspond to novel and meaningful images. It turns out that score smoothing is crucial for diffusion models to achieve this.

Remarkably, in multi-dimensional settings, the effect of score smoothing manifests in a direction-dependent manner. Along directions that are parallel (or “tangential”) to the hidden data manifold, it produces a similar slowing-down effect as in the 1-D scenario. However, along directions pointing towards the manifold, the “perfect” score function is already relatively smooth (in fact, just a straight line if the manifold is flat), and further smoothing does not make much difference.

Hence, instead of applying brakes to the particles’ flow in every direction (which would stall them in the noisy empty space and result in the final images being blurry), score smoothing does not slow down their movement toward the manifold, but only reduces their tendency to collapse towards the training data along the tangential directions. In this way, the model achieves a balance between quality and novelty: the images are both realistic looking (because they successfully reached the meaningful data manifold) and new (because they settled into the blank spaces between the original training data points).