  Denoising Diffusion Models (DDMs) have emerged as a strong competitor to
Generative Adversarial Networks (GANs). However, despite their widespread use
in image synthesis and editing applications, their latent space is still not as
well understood. Recently, a semantic latent space for DDMs, coined
`$h$-space', was shown to facilitate semantic image editing in a way
reminiscent of GANs. The $h$-space is comprised of the bottleneck activations
in the DDM's denoiser across all timesteps of the diffusion process. In this
paper, we explore the properties of h-space and propose several novel methods
for finding meaningful semantic directions within it. We start by studying
unsupervised methods for revealing interpretable semantic directions in
pretrained DDMs. Specifically, we show that global latent directions emerge as
the principal components in the latent space. Additionally, we provide a novel
method for discovering image-specific semantic directions by spectral analysis
of the Jacobian of the denoiser w.r.t. the latent code. Next, we extend the
analysis by finding directions in a supervised fashion in unconditional DDMs.
We demonstrate how such directions can be found by relying on either a labeled
data set of real images or by annotating generated samples with a
domain-specific attribute classifier. We further show how to semantically
disentangle the found direction by simple linear projection. Our approaches are
applicable without requiring any architectural modifications, text-based
guidance, CLIP-based optimization, or model fine-tuning.
