Matteo Benati, Alessandro Londei, Denise Lanzieri, Raffaello Mastromarino, Vittorio Loreto · 2026 · arXiv
DOI: 10.48550/arXiv.2610.30009 · open-access PDF
Training generative models on synthetic data causes loss of diversity and a decrease in performance, a phenomenon known as model collapse. We study this degeneration spectrally in a controlled Markov setting. The eigenvalue spectrum of the learned transition operator offers a direct view of the process, since the subdominant eigenvalues set how fast the chain mixes and therefore how much of the state space it continues to explore. We find that sampling at temperature $T_s < 1$ acts on the operator like repeated sharpening, pushing it toward a deterministic system: the subdominant eigenvalue
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