octaves

Noise covers things that feel unrelated — a fluctuation measured in someone's walking, an algorithm written to make a convincing cloud, the grey static a diffusion model starts from, the levelling described in entropy. The word is not being stretched. Four independent axes run through it, and most of the apparent confusion is those four coming apart.

The first is found or made. Pink noise is something you find: measure a system, subtract the average, and read what is left. Perlin noise is something you make: an algorithm you write and feed to a system that needs a surface. That is not a difference in the noise but in whether you are analysing or synthesising — the asymmetry Braitenberg's vehicles names as its law, since it is cheap to build something that produces convincing structure and expensive to read the structure back out. Found noise is uphill; made noise is downhill.

The second axis is the one that actually settles the family question: correlated or not. White noise is the outlier here, flat and uncorrelated, with no relation between one scale and the next. Pink noise is defined by having structure at every scale, its power falling off as one over frequency — which is what the long memory in the walking case amounts to. And the standard way of building Perlin noise, summing octaves each at half the amplitude and double the frequency of the last, produces a spectrum that falls off in approximately the same way. So the resemblance between a measured fluctuation and a written algorithm is not poetic. It is spectral, and it is engineered on purpose.

The third axis is distribution rather than spectrum, and it is where Gaussian noise belongs — because Gaussian is not a fourth colour alongside white and pink. White and pink describe how a signal is spread across frequencies. Gaussian describes how its values are spread: a bell curve, mostly small excursions with occasional large ones, symmetric about zero. The two properties are independent, so there is white Gaussian noise and pink Gaussian noise both, and asking whether Gaussian is comparable to pink is close to asking whether loud is comparable to red.

Two consequences make this more than a technicality. A bell curve is the shape that many independent contributions take when they are added together and none dominates — which means a Gaussian is the signature of aggregation itself, and the same operation as the statistical average seen from a different side. That is the bridge between this family and averaging toward dominance: whatever was most numerous ends up at the centre, in both. And among all distributions with a given spread, the Gaussian is the one carrying maximum entropy — the most levelled a signal can be at a given power. So the grey static a diffusion model starts from is not merely random. It is the most undifferentiated state available, which makes denoising a literal reversal of the levelling.

The fourth axis is the valency the whole set already carries: levelling, as in entropy, against generative, as in order from noise. These are independent of the other three. A diffusion seed is made, uncorrelated, Gaussian, and generative. Physical entropy is found, uncorrelated, Gaussian, and destructive — thermal fluctuation in a resistor is both. Pink noise is found, correlated, and diagnostic. Which is worth noticing at the bench: the jitter a sensor contributes under transduction and the seed a model starts from are the same kind of thing arriving from opposite directions.

Which gives early stop a reason rather than a description. Denoising is the progressive installation of correlation — coarse structure first, fine structure last. Halting a fifth of the way in does not leave a partial picture. It leaves a field correlated at large and middle scales and not yet at small ones.