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The increasing presence of AI-generated content on the internet raises a critical question: What happens when generative machine learning models are pretrained on web-scale datasets containing data created by earlier models? Some authors prophesy $\textit{model collapse}$ under a "$\textit{replace}$" scenario: a sequence of models, the first trained with real data and each later one trained only on synthetic data from its preceding model. In this scenario, models successively degrade. Others see collapse as easily avoidable; in an "$\textit{accumulate}$' scenario, a sequence of models is trained, but each training uses all real and synthetic data generated so far. In this work, we deepen and extend the study of these contrasting scenarios. First, collapse versus avoidance of collapse is studied by comparing the replace and accumulate scenarios on each of three prominent generative modeling settings; we find the same contrast emerges in all three settings. Second, we study a compromise scenario; the available data remains the same as in the accumulate scenario -- but unlike $\textit{accumulate}$ and like $\textit{replace}$, each model is trained using a fixed compute budget; we demonstrate that model test loss on real data is larger than in the $\textit{accumulate}$ scenario, but apparently plateaus, unlike the divergence seen with $\textit{replace}$. Third, we study the relative importance of cardinality and proportion of real data for avoiding model collapse. Surprisingly, we find a non-trivial interaction between real and synthetic data, where the value of synthetic data for reducing test loss depends on the absolute quantity of real data. Our insights are particularly important when forecasting whether future frontier generative models will collapse or thrive, and our results open avenues for empirically and mathematically studying the context-dependent value of synthetic data.

Diffusion models have a tendency to exactly replicate their training data, especially when trained on small datasets. Most prior work has sought to mitigate this problem by imposing differential privacy constraints or masking parts of the training data, resulting in a notable substantial decrease in image quality. We present CPSample, a method that modifies the sampling process to prevent training data replication while preserving image quality. CPSample utilizes a classifier that is trained to overfit on random binary labels attached to the training data. CPSample then uses classifier guidance to steer the generation process away from the set of points that can be classified with high certainty, a set that includes the training data. CPSample achieves FID scores of 4.97 and 2.97 on CIFAR-10 and CelebA-64, respectively, without producing exact replicates of the training data. Unlike prior methods intended to guard the training images, CPSample only requires training a classifier rather than retraining a diffusion model, which is computationally cheaper. Moreover, our technique provides diffusion models with greater robustness against membership inference attacks, wherein an adversary attempts to discern which images were in the model's training dataset. We show that CPSample behaves like a built-in rejection sampler, and we demonstrate its capabilities to prevent mode collapse in Stable Diffusion.

One of the major outstanding conjectures in the study of chromatic symmetric functions (CSF's) states that trees are uniquely determined by their CSF's. Though verified on graphs of order up to twenty-nine, this result has been proved only for certain subclasses of trees. Using the definition of the CSF that emerges via the Frobenius character map applied to $\mathbb{C}[S_n]$, we offer new algebraic proofs of several results about the CSF's of trees. Additionally, we prove that a "parent function" of the CSF defined in the group ring of $S_n$ can uniquely determine trees, providing further support for Stanley's conjecture., Comment: 10 pages; small typos corrected

We study the prime numbers that lie in Beatty sequences of the form $\lfloor \alpha n + \beta \rfloor$ and have prescribed algebraic splitting conditions. We prove that the density of primes in both a fixed Beatty sequence and a Chebotarev class of some Galois extension is precisely the product of the densities $\alpha^{-1}\cdot\frac{|C|}{|G|}$. Moreover, we show that the primes in the intersection of these sets satisfy a Bombieri--Vinogradov type theorem. This allows us to prove the existence of bounded gaps for such primes. As a final application, we prove a common generalization of the aforementioned bounded gaps result and the Green--Tao theorem., Comment: 20 pages

We study the chromatic symmetric functions of graph classes related to spiders, namely generalized spider graphs (line graphs of spiders), and what we call horseshoe crab graphs. We show that no two generalized spiders have the same chromatic symmetric function, thereby extending the work of Martin, Morin and Wagner. Additionally, we establish that a subclass of generalized spiders, which we call generalized nets, has no e-positive members, providing a more general counterexample to the necessity of the claw-free condition. We use yet another class of generalized spiders to construct a counterexample to a problem involving the $e$-positivity of claw-free, P4-sparse graphs, showing that Tsujie's result on the e-positivity of claw-free, P4-free graphs cannot be extended to graphs in this set. Finally, we investigate the e-positivity of another type of graphs, the horseshoe crab graphs (a class of unit interval graphs), and prove the positivity of all but one of the coefficients. This has close connections to the work of Gebhard and Sagan and Cho and Huh., Comment: 20 pages; corrected typo