objective Trace gradients through sampling
Instead of sampling `z ~ N(μ, σ²)` directly (which blocks gradients), we reparameterise as `z = μ + σ · ε` where `ε ~ N(0, I)`. Kingma & Welling 2014 showed this makes the sampling differentiable w.r.t. `μ` and `σ`.
import torch
def reparameterise(mu, log_var):
sigma = torch.exp(0.5 * log_var)
eps = torch.randn_like(sigma)
return mu + sigma * eps
mu = torch.zeros(1, 32)
logvar = torch.zeros(1, 32)
z = reparameterise(mu, logvar)
print(z.shape, z.mean().item())torch.Size([1, 32]) -0.018
What is the role of `ε` in the reparameterisation trick?
$ python train.py --epochs 3 --lr 1e-3 Epoch 1/3 · loss 184.2 Epoch 2/3 · loss 168.7 Epoch 3/3 · loss 156.1 Checkpoint → ./runs/vae_mnist.pt
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Last updated: June 2026