Generative collapse does not imply discriminative collapse

Recursive training on synthetic data can lead to model collapse, in which learned generative distributions progressively lose diversity or degenerate across training generations. In this paper, we ask whether such generative degeneration necessarily deteriorates the downstream task performance. We distinguish generative collapse, meaning collapse of the learned data distribution, from discriminative collapse, referring to degradation in performance on downstream tasks such as classification or regression. We theoretically investigate this distinction in a solvable setting: the iterative maximum likelihood estimation (MLE) for the symmetric binary Gaussian mixture model under the discard workflow, where each generation is trained only on synthetic samples from the previously fitted model. In line with a widely known covariance collapse phenomena under the multivariate Gaussian distribution, the fitted covariance matrix also vanishes to zero almost surely, so the trained generative model becomes degenerate under the symmetric binary GMM. Our main finding is that this generative collapse does not necessarily imply collapse of the performance on the downstream classification task. Although the covariance collapses, the fitted mean vector converges almost surely to a non-degenerate random vector that retains directional information about the population signal. Consequently, the regularized plug-in linear discriminant analysis (LDA) classifier has the limiting expected risk strictly smaller than $1/2$, i.e., it is superior to random guessing. Moreover, in the high-SNR regime, its limiting expected risk approaches to the Bayes risk at a quadratic rate keeping the condition number of the ground truth covariance matrix constant. We further show that within the large-sample regime as $n\to \infty$, the terminal mean vector converges weakly to the multivariate $t$-distribution with degrees of freedom $4$, centered at the true mean vector. These results show that generative collapse does not always lead to discriminative collapse.