Noise, score, and velocity: transform the target and its error together

Use a²+σ²=1 and define v=aε−σx_0. Together with x_t=ax_0+σε, this is an orthogonal two-coordinate rotation applied independently to every data coordinate. Its inverse recovers x_0 and ε without dividing by a or sigma. This target is called velocity in the cited diffusion convention; it is not automatically a physical velocity or the derivative of every sampler trajectory. Other interpolations and flow-matching conventions can use different signs, coefficients, and targets.

A predicted v gives corresponding clean and noise predictions through the inverse identities. A noise prediction instead reconstructs x_0 by dividing x_t−σε_hat by a, requiring a>0. A clean prediction reconstructs noise by dividing x_t−ax_0_hat by sigma, requiring sigma>0. These divisions can amplify finite errors near an endpoint. A scheduler's prediction_type must agree with the network's training target; changing a configuration label without transforming the output does not convert a trained model.

For the forward Gaussian conditional on a known clean sample, differentiating its log density with respect to x_t gives −ε/σ. Denoising score matching trains against that known noisy target. Under squared error and appropriate regularity, its population optimum given x_t is the conditional expectation of that target, which equals the score of the perturbed data marginal. The individual target is not already the unknown marginal score for that particular clean/noise draw. With external conditioning, the corresponding marginal is conditioned on that information.

Holding x_t and time fixed, a velocity prediction error delta_v produces noise error a delta_v and clean error −σ delta_v. The score error obtained from the noise prediction is −a delta_v/σ. Squaring introduces coefficient factors, not just a change of variable name. These identities compare predictions tied by the stated conversion. They do not say two independently parameterized finite networks have identical gradients, initialization, capacity, or training trajectories, even if their population-optimal predictions are related.

Create a small tensor with fixed clean values, Gaussian draws, and nondegenerate coefficients. Check round-trip conversions and the expected loss factors in a wide reference dtype before testing mixed precision. Add near-endpoint cases separately and define allowed tolerances, because division-based conversions can lose accuracy. Verify coefficient broadcasting and target selection across the batch. A low scalar loss under the wrong target can be misleading if its magnitude or timestep weighting differs from the intended objective.

The matrix lab offers a separate sensitivity example. Reset, set a=1 and b=0, predict condition number one, then Play or Step. Separately Reset, set a=0.25 and b=0, predict singular values one and 0.25 and condition number four, then Play or Step. Readouts describe the selected final triangular matrix while playback interpolates geometry. This is not the velocity rotation or a diffusion target converter. Use its explicit small-scale direction to motivate checking error amplification in the actual division-based formulas.