Encode to Latent with the Scaling Factor

1. Background Knowledge

In Latent Diffusion Models (such as Stable Diffusion), images are not denoised directly in pixel space. Instead, a Variational Autoencoder (VAE) compresses the image into a lower-dimensional latent space. This reduces computational cost while preserving semantic structure. The VAE encoder produces a raw latent vector, $z_{\text{raw}}$, which typically has a distribution that does not match the assumptions of the diffusion noise schedule.

The diffusion process assumes that the data being denoised has approximately unit variance (standard deviation of 1). If the raw latents have a different scale, the noise schedule (which adds Gaussian noise with a fixed variance) will be mismatched, leading to poor generation quality. To fix this, Stable Diffusion applies a fixed scaling factor $s$ to the raw latents:

$z = s \cdot z_{\text{raw}}$

For Stable Diffusion 1.x and 2.x, this factor is empirically set to $s = 0.18215$. This scaling ensures the latent distribution is centered near zero with unit standard deviation, aligning with the diffusion model’s training assumptions.

2. Algorithm Approach

This problem is a straightforward element-wise scalar multiplication followed by rounding. There is no complex algorithmic pattern here; the core operation is:

1. Multiply each element of the input list by the scalar scale.
2. Round each result to 6 decimal places.
3. Return the resulting list.

This is an $O(n)$ operation where $n$ is the length of the latent vector.

3. Step-by-Step Strategy

• Step 1: Iterate over the input list. Loop through each value in z_raw.

• Step 2: Apply the scaling factor. For each value $z_i$, compute $z_i \cdot \text{scale}$.