Before diffusion models generalize, they grow spurious attractors absent from training data but repeated in samples

As the training set grows from small to large, diffusion models pass through an intermediate stage in which generated samples are neither copies of training images nor fresh samples on a flat manifold, but sit at the bottom of newly formed attractors in the energy landscape. Earlier work on the memorization-to-generalization transition approached it from the generalization side, treating memorization as a small side effect alleviated by greatly increasing training data, leaving the intermediate regime unexamined. On MNIST, FASHION-MNIST, CIFAR-10 and downsampled LSUN-CHURCH, 38 DDPM models were trained per dataset across data sizes, from a smallest model with 2 training points to a largest trained on the full set, each with a synthetic set 4 times its training set.

These intermediate samples are classed as spurious: they have exact duplicates in the synthetic set, indicating a well-defined basin of attraction, yet no counterpart in the training set. Spurious states were historically treated in associative memory as detrimental artifacts hindering retrieval; here they are read as the first signs of generative capability. Thresholds taken from two bimodal distance histograms, to the nearest training neighbor and to the nearest synthetic neighbor, classify each generated sample as memorized, spurious or generalized; the spurious fraction rises and then falls with training set size, peaking at the boundary between memorization and generalization.

The three sample types separate by basin volume and energy curvature: memorized samples have the largest log-volume of attraction, spurious samples come next, and generalized samples have near-zero volume. Rather than a simple memorization-versus-generalization split judged by copy rates alone, this assigns a geometric quantity to each of the three states. Basin volume is computed from the radius set by the latest time at which the reverse process still recovers a target sample; singular value decomposition of the score Jacobian separates memorized, spurious and generalized samples in the smallest-singular-value region on MNIST and LSUN-CHURCH, and curvature at training points drops on average as the training set grows.

A two-dimensional toy model makes the energy landscape evolution directly visible: with training points on the unit circle, K=2 gives memorization, K=9 shows the first spurious states, and K=1000 leaves low-energy states on a flat manifold. High-dimensional natural datasets make the energy hard to visualize, so this controlled setting displays the memorization, spurious and generalization phases separately. The model uses a variance-exploding stochastic differential equation diffusion pipeline and is compared against the analytic energy given by a modified Bessel function in the limit of infinite K.

The conclusions rest on diffusion models trained with standard score matching, covering MNIST, FASHION-MNIST, CIFAR-10 and downsampled LSUN-CHURCH plus a two-dimensional toy model on the unit circle; the Stable Diffusion analysis involves only a small number of prompts and generated samples and is a qualitative comparison. The authors suggest that if spurious states become abundant during training one could stop training or add data, and that regularizers penalizing high curvature could be added to the score network or loss, but these remain future directions.

The theory assumes the score network is large enough and trained long enough to learn the empirical distribution of noised data, with generation stopped at a small non-zero time; practical networks are finite, and a feed-forward network is not guaranteed to output the gradient of a scalar energy. The authors state they do not prove spurious states causally drive generalization, only that the convergence of observations makes this a plausible interpretation; curvature spectra do not separate cleanly at every training set size. Effects of training time and early stopping on spurious states are left unexamined.