Forward noising: state the normalization before quoting an SNR

This module uses x_t=a_t x_0+σ_t ε with independent standard Gaussian noise and a_t²+σ_t²=1. The name variance-preserving refers to the coefficient construction; a data coordinate with unit variance then retains unit marginal variance. Conditional on a fixed clean sample, the variance is only σ_t². If the clean data has another variance or nonzero mean, its marginal moments change accordingly. Images or autoencoder latents therefore need a recorded normalization rather than an assumed unit-scale convention.

For discrete DDPM, each one-step variance beta defines alpha=1−beta, while the cumulative product of alphas determines the direct marginal coefficients. Sampling that marginal avoids simulating all previous forward steps for every training example. It is still a statement about a specific process and timestep index. A scheduler using another continuous-time SDE, a different signal scale, or another interpolation requires its own equations. Do not combine its sigmas with this module's target formulas without checking the conversion.

Under independent zero-mean unit-variance clean coordinates and noise, signal variance is a_t² and added noise variance is σ_t². Their ratio is the commonly used coefficient SNR. For a coordinate with clean variance v, the actual signal-to-added-noise variance ratio is a_t²v/σ_t². Rescaling a latent representation therefore changes this ratio unless the schedule is adjusted. Record whether a reported SNR is the coefficient ratio, a measured data-scale quantity, or a log-SNR used as a network input.

At zero added noise, the ratio diverges and score targets involving division by sigma are undefined under their ordinary formulas. At zero signal, converting a predicted noise value into a clean sample by dividing by a is undefined. Practical schedules and endpoint handling must state which values are trained and which are only limiting cases. More diffusion timesteps do not automatically improve learning if their noise levels, sampling distribution, and weights are changed at the same time.

Check a tiny batch against hand-computed coefficients, then estimate conditional means and variances over repeated noise draws. Verify tensor broadcasting, timestep indexing, dtype, normalization, and that the sampled noise used in the target is the intended one. A debug run should preserve its clean samples, noise, and timesteps so two implementations can compare identical inputs. A plot that looks noisier at later indices is not enough to establish the exact training distribution or target alignment.

The function lab isolates the square turning noise amplitude into variance. Reset, set a=1 and x=0.60 in y=ax², predict y=0.36, then Play or Step. Separately Reset, set a=1 and x=0.80, predict y=0.64, then Play or Step. Here the graph's x is an amplitude proxy, not diffusion time, and its coefficient is unrelated to the forward signal coefficient a_t. It contains no Gaussian sampler or image. Use the squared values with the separate VP equation to compute the matching signal coefficients.