The maths of diffusion and VAEs

Implement the formulas behind the forward diffusion process and the VAE loss.

• linear_betas(T, start=1e-4, end=0.02): T noise variances evenly spaced from start to end.
• alpha_bar(betas): the running product ${\bar{\alpha }}_t={\prod }_{s\le t}(1-{\beta }_s)$ (an array the same length as betas).
• q_sample(x0, t, eps, abar): the noised image $\sqrt{{\bar{\alpha }}_t}{x}_0+\sqrt{1-{\bar{\alpha }}_t}ϵ$ using abar[t].
• vae_kl(mu, logvar): $\frac{1}{2}\sum ({\mu }^2+e^{\text{logvar}}-1-\text{logvar})$ over all elements.